The capacitance (dielectric) technique has been widely used to estimate soil volumetric water content (θ) [3]. Capacitance sensors induce an alternating electric field in the surrounding medium. The total complex impedance is obtained by quantifying the voltage and the current induced by the electric field on the sensor electrodes. The impedance is related to the complex permittivity (or dielectric constant; ε_r) of the surrounding medium. The volume of the induced electric field depends mainly on the size and shape of the sensor electrodes. Moreover, the electric field decays rapidly, being inversely proportional to the square of the distance. Topp et al.[4] noticed a strict correlation between ε_r measured by time domain reflectometry (TDR) and soil water content. They therefore proposed an empirical third-degree polynomial in ε_r to calculate θ. The complex permittivity of the soil measured by dielectric sensors is the sum of soil real (ε’) and imaginary (ε″) permittivity (dielectric loss): (1) ε r = ε ′ − j × ε ″where j² = −1. The value of θ is related to ε’ only. On the other hand, ε″ changes according to soil salinity, soil temperature (T), and the operating frequency of the sensor [5–10]. Especially in low-cost sensors working at low frequencies (<1 GHz), the contribution of ε″ in saline soils cannot be ignored [3,11,12]. It is therefore essential to consider the influence of ε″ in ε_r measurements in order to gain correct θ estimations.

The determination of the pore-water electrical conductivity is a difficult task as it cannot be directly related to any sensor output. Typically sensors measure soil bulk (or apparent) electrical conductivity (EC_a), which is the combination of the contributions of the three phases constituting soils: solid, water and air [9,13]. According to Corwin [14], three pathways of current flow contribute to the EC_a measurement: current through the pore water solution (the liquid phase pathway); current through exchange complexes on the surface of soil colloids (the soil-liquid phase pathway); and current through the soil particles that are in direct contact (the solid pathway). EC_a can be estimated from ε_r readings [15] or from the electrical resistance that soil opposes to an alternating electric current [13,14]. EC_p and EC_a are strictly correlated, indeed an increase of ions in the matrix solution leads to an increase of EC_a values [8,16,17].

Several models to estimate EC_p from EC_a have been developed in the last sixty years, based on empirical relations as well as on theoretical assumptions. Models are usually based on the empirical relationship between EC_a and θ at constant EC_p values, where the magnitude of EC_a varies according to the tortuosity of the electrical current paths (depending on soil texture, density and particle geometry, particle pore distribution, and organic matter content). Tortuosity can be expressed in terms of a soil transmission factor (π) [16,18,19] or soil-type-related parameters [20–22].

Recent development of low-cost multi-sensor probes could make such EC_p models implementable for continuous monitoring purposes. However, since most of the EC_p models are calibrated in limited soil conditions [9,23–25], new relationships between variables and soil properties must be defined to extend their applicability to a wider range of soils.

The general aim of this study was to calibrate a multi-sensor probe for monitoring soil volumetric water content and soil water electrical conductivity in a heterogeneous saline coastal area. The specific objectives were: (i) to develop a procedure to simultaneously calibrate θ and EC_p; (ii) to test different models for EC_p; and (iii) to develop general functions to extend EC_p model application to a wide range of soils, even in critical saline conditions.

The sensor used in this experiment was an ECH₂O-5TE probe (hereafter simply referred to as 5TE). 5TE is a multifunction sensor measuring ε_r, EC_a, and T (Decagon Devices Inc., Pullman, WA, USA). A detailed description of the 5TE can be found in Bogena et al.[26] and Campbell and Greenway [27]. The probe is a fork-type sensor (0.1 m in length, 0.032 m in height). Two of the three tines host the dielectric sensor. The capacitance sensor supplies a 70 MHz electromagnetic wave to the prongs that charge according to the dielectric of the soil surrounding the sensor. The reference soil volume is ca. 3 × 10⁻⁴·m³. A charge is consequently stored in the prongs and it is proportional to the soil dielectric. Previous versions of dielectric sensors by Decagon Devices operate at lower frequencies (e.g., ECHO10 probe, 5 MHz). The increase of operating frequency has led to a higher salinity tolerance [6,8,12]. In fact ε_r measurement with 5TE should not be affected by soil salinity up to EC_e values of 10 dS·m⁻¹[28].

The bulk electrical conductivity is measured with a two-sensor array. The array consists of two screws placed on two of the sensor tines. An alternating electrical current is applied on the two screws and the resistance between them is measured. The sensor measures electrical conductivity up to 23.1 dS m⁻¹ with 10% accuracy; however a user calibration is suggested above 7 dS·m⁻¹. Temperature is measured with a surface-mounted thermistor reading the temperature on the surface of one of the prongs.

Soil samples from a coastal farmland affected by saltwater intrusion [29,30] were cored for the calibration of the 5TE probe. The site is located at Ca’ Bianca, Chioggia (12°13′55.218″E; 45°10′57.862″N), just south of the Venice Lagoon, North-Eastern Italy. The area has high spatial variability in soil characteristics due to its deltaic origins (Figure 1).

Three sampling locations were chosen in the basin (sites A, B, and C, Figure 1). At sites A and B both topsoil (0 to 0.4 m depth) and subsoil (0.4 to 0.8 m depth) were collected, while only the topsoil was cored at site C since the profile is uniform. The main physical and chemical properties of the samples were characterized. Soil texture was determined with a laser particle size analyzer (Mastersizer 2000, Malvern Instruments Ltd., Great Malvern, UK). Soil total carbon content and soil organic carbon (SOC) content were analyzed with a Vario Macro Cube CNS analyzer (Elementar Analysensysteme GmbH, Hanau, Germany). Cation exchange capacity (CEC) was measured at a pH value of 8.2 according to the BaCl extraction method [31]. Soil pH was measured with a 1:2 soil to water ratio with a pH-meter (S47K, Mettler Toledo, Greifensee, Switzerland). Particle density (ρ_r) was measured with an ethanol pycnometer [32]. Bulk density (ρ_b) was determined from undisturbed core samples. EC_e was measured according to Rhoades et al.[1].

Soil samples show high variability in sand (from 174.7 to 905.2 g·kg⁻¹), organic carbon content (from 15.4 to 147.8 g·kg⁻¹), and EC_e values (from 0.61 to 6.38 dS·m⁻¹). Five soil types were selected: a sandy soil with low SOC content and low EC_e, a silty-clay-loam with low SOC content and high EC_e, two loam and one clay-loam with medium-high SOC content. Main soil properties are listed in Table 1.

The 5TE probe was used in a mixture of soil (preliminarily air-dried and sifted at 2 mm) and saline solution (54.92% Cl⁻; 30.82% Na⁺; 7.68% SO₄²⁻; 3.81% Mg²⁺; 1.21% Ca²⁺; 1.12% K⁺; 0.44%NaHCO₄) to reproduce saline groundwater of the experimental site [33]. Soil samples were moistened to a relative saturation (S) of about 0, 0.35, 0.75, and 1.00 with a saline solution of 0, 5, 10, and 15 dS·m⁻¹ (at 25 °C). The mixtures were prepared in a plastic container and then sealed and kept in a dark place at constant temperature 22 ± 1 °C for 48 hours. The soil was then packed uniformly in a 6 × 10⁻⁴·m³ beaker to reproduce the field bulk density. Output values for ε_r, EC_a, and T were recorded by a datalogger (Em50, Decagon Devices) connected to the 5TE probe.

Electrical conductivity of the wetting solution (EC_w) differs from the electrical conductivity of the pore-water (EC_p) [21]. Pore-water solution was extracted from a portion of the soil sample by vacuum displacement [34] at −90 kPa and EC_p was measured with a S47K conductivity meter. EC_e was then measured on the remaining soil sample. Water content was determined gravimetrically (at 105 °C for 24 hours). Measures were replicated 3 times.

A three-step procedure was implemented to calibrate the sensor output for the collected samples: (1) model calibration to convert ε_r and EC_a readings to θ or EC_p; (2) comparison and selection of the best models; (3) simultaneous calibration of the selected models for θ and EC_p and evaluation of their robustness by applying a bootstrap procedure.

Dependence of 5TE on bulk electrical conductivity was observed to be similar in all the tested soil samples. ε_r readings were greatly affected by EC_a: especially for high θ values, a small increase in EC_w significantly raised the dielectric output of the probe, indicating that dielectric readings carried out in highly conductive media must be corrected. This finding confirms the results by Rosenbaum et al.[5] on the same probe and by Saito et al.[8] on other Decagon dielectric probes operating at lower frequencies. An example of the non-linear response of ε_r at different EC_w and θ values is presented in Figure 2(a). Starting from a relative saturation of 0.75, the response of the probe significantly diverged at salinity solution with EC_w > 10 dS m⁻¹. Figure 2(b) evidences also the direct effect of the EC_w on EC_a readings and how the effect was amplified at higher water content. This observation, confirmed by Schwank et al.[11] and Rosenbaum et al.[5], suggests investigating the effect of EC_a on θ estimation.

Between the tested θ models, Equation (4) showed the best performances, with an Akaike weight W_AIC close to 1 (Table 2).

Parameters a and b of the logarithmic model were found to be significantly correlated with the EC_a values at different water contents. Therefore the logarithmic model was reformulated as: (14) θ = ( a ′ + a ″ × E C a ) + ( b ′ + b ″ × E C a ) × ln ( ε r )where a′, a″, b’, and b″ are empirical parameters. Calibration of Equation (14) highlighted a strong correlation between the terms a′ × EC_a+ a″ and b’× EC_a+ b″. Consequently this latter term was assumed equal to q× (a′×EC_a+ a″), where q is a proportionality constant. Equation (14) could thus be reformulated as: (15) θ = ( a ′ + a ″ × E C a ) × ( 1 + q × ln ( ε r ) )

Equations (14) and (15) were compared with the AIC test. A W_AIC = 0.99 was obtained for Equation (15), indicating that this formulation of the logarithmic model is to be preferred over Equation (14), mainly for the reduced number of parameters.

To identify a “general” equation, q was set as a constant (=−0.766), whereas parameters a′ and a″ were related to soil properties (Table 3). Parameters a′ and a″ were estimated according to the following empirical equations: (16) a ′ = − 0.352 − 0.006 × S O C ( % ) (17) a ″ = 0.020 − 0.009 × clay ( % ) sand ( % )

Equation (15) allows correcting of the effect of dielectric losses due to the high electrical conductivity of the medium [3] due to high organic carbon content, salinity, and clay/sand ratio. The RMSE of Equation (15) was 0.038 m³·m⁻³.

The parameters of models (5), (6), (8), and (11) showed significant correlations with soil properties (Table 3).

The parameter l by Malicki and Walczak was confirmed to be mainly correlated to sand content. The calibrated parameters for Equation (5) are: EC_a0 = 0.06 dS·m⁻¹; ε_r0 = 7.1; l = 0.012 + 10⁻⁶× sand(%) yielding RMSE = 2.52 dS·m⁻¹. In the original paper by Malicki and Walczak l varied from 0.0083 to 0.0127 while in this experiment the range was narrower, from 0.0117 to 0.0124.

The ε_{(ECa = 0)} parameter by Hilhorst was expressed as a function of soil organic carbon content: (18) ε E C = 0 = 4.851 + 0.203 × S O C ( % )

According to Equation (18), ε_{ECa = 0} ranged from 5.16 to 7.85, values close to the interval found by Hilhorst (from 3.76 to 7.6 in soils and synthetic media). The calibrated model yielded RMSE = 2.34 dS m⁻¹.

Concerning the Rhoades and Archie models, the term EC_s was neglected as the 5TE probe registered EC_a = 0 in dry soil conditions. Please note that other Authors [41] demonstrated that EC_s could assume a certain magnitude.

In contrast to Equation (9) by Rhoades et al.[16], π was found to be uncorrelated with soil water content. Nevertheless, π showed a linear correlation with soil porosity (π = e + f × ϕ), with e and f depending on CaCO₃ as follows: (19) π = ( 0.129 × CaCO 3 ( % ) − 1.779 ) + ( − 0.232 × CaCO 3 ( % ) + 3.989 ) × ϕ

In the tested samples π ranged from 0.22 to 0.71, whereas Rhoades et al.[16] found a variation from 0.01 to 0.6. The inverse correlation of CaCO₃ with the tortuosity factor evidenced in the Ca’ Bianca soils can be explained by the fact that here a low CaCO₃ content corresponds to high clay and SOC percentages. Indeed, the higher clay and SOC contents (more complicated geometric arrangement), the higher is soil tortuosity [41]. The “general” formulation of the Rhoades model provided RMSE = 0.90 dS m⁻¹.

Several formulations were attempted for Archie in order to decrease the number of parameters related to soil properties. Here, the parameters k, m, and n were alternately fixed and kept independent from the soil type. The formulation with k = 0.487 provided the best fitting according to the AIC test. With fixed k, n showed a significant correlation with sand content: (20) n = − 0.669 + 0.035 × sand ( % )

It is worth noticing that with higher sand contents (Figure 3(a)) n ≅ 2.5, which is close to n values suggested for sandy media [13]. As shown in Table 3, n decreases with increasing clay values. For given S and EC_a values, it is clearly derived from Equation (10) that the smaller the n the higher is EC_p, i.e., with a large percentage of clay the influence of “the liquid phase pathway” on the EC_a reading is reduced [14,41]. A non-linear relationship was detected between m and soil organic carbon (Figure 3(b)): (21) m = − 0.018 × S O C ( % ) + 4.350 × S O C ( % ) S O C ( % ) + 0.966

Values of m between 2.65 and 3.82 were derived. As reported by Archie, m becomes larger as the permeability of the porous medium decreases (increasing tortuosity). As shown in Figure 3(b) the magnitude of m rises with SOC. High organic contents decrease soil bulk density, possibly increasing soil tortuosity [46]. Archie calibration returned RMSE = 0.65 dS m⁻¹.

Comparison between the RMSE values computed for the four EC_p models showed that the “general” formulation of Archie provided the best estimates. Archie also had the highest W_AIC (∼1.00). For the Malicki and Walczak, Hilhorst, and Rhoades models the W_AIC were close to zero.

These results are also confirmed by the linear regressions between measured and estimated EC_p (Figure 4). As displayed in this figure the models by Malicki and Walczak, and by Hilhorst did not show a good fitting, especially at high EC_p values, as already observed by [23,25].

The different performances of the four models at various salinity ranges were tested resampling observed and estimated EC_p 2,000 times, to compute average RMSEs and their confidence intervals at p = 0.05 as previously done by Giardini et al.[47]. The selected ranges were: (a) the 0–3 dS·m⁻¹ and >3 dS·m⁻¹; and (b) the 0–10 dS·m⁻¹ and >10 dS·m⁻¹.

At low EC_p range (i.e., EC_p < 3 dS·m⁻¹) Rhoades showed the smallest RMSE (0.57 dS·m⁻¹), nevertheless its performance was not significantly different from those by Hilhorst (RMSE = 0.93 dS·m⁻¹) and Archie (RMSE = 0.72 dS·m⁻¹). On the contrary, the model by Malicki and Walczak provided significantly higher errors (RMSE = 1.69 dS·m⁻¹).

Above 3 dS·m⁻¹, the models by Malicki and Walczak and by Hilhorst significantly differentiated from the other two. In fact they generally overestimated EC_p in the range from 3 to 10 dS·m⁻¹ with RMSE equal to 2.16 and 1.43 dS·m⁻¹, respectively. On the other hand they underestimated EC_p when the pore-water was very conductive (i.e., EC_p > 10 dS·m⁻¹), with RMSE = 3.28 dS·m⁻¹ and RMSE = 3.83 dS·m⁻¹, respectively.

In their work, Malicki and Walczak used TDR probes at fairly high frequencies, reducing the influence of EC_a on ε_r. Moreover, their study was conducted using a wetting solution with a maximum conductivity of 11.7 dS·m⁻¹. In the present work, calibrating the Malicki and Walczak model only for EC_p < 10 dS·m⁻¹ would provide satisfactory estimations (RMSE = 1.00 dS·m⁻¹). Moreover, the metrics of fitting regression would have shown a slope and intercept of 0.837 and 0.680, yielding very similar results to those obtained by Malicki and Walczak in their work. With some limitations, the model by Malicki and Walczak might therefore be used in capacitance applications as well as TDR [21] and frequency-domain reflectometry [10].

Hilhorst validated his model in a much lower EC_p range than the one used in this work. Hilhorst actually indicated the validity upper bound for the probe used in his work as 3 dS·m⁻¹. Indeed, in the present study the model showed good performances in the 0–3 dS·m⁻¹ range. Moreover, calibrating the model for EC_p <10 dS·m⁻¹ would suitably yield a RMSE of 0.68 dS·m⁻¹ with an observed-estimated relationship having a slope and an intercept of 0.957 and 0.127, respectively. Most likely, the higher operating frequency of 5TE compared to the capacitive probe used by Hilhorst (i.e., 30 MHz) could have increased the range of model validity. However, as stated by Hilhorst, the model assumptions cease to be valid at higher salt concentrations as ε_p significantly deviates from that of free water (Equation (7)). From the experiment presented here this limit seems to be EC_p ∼ 10 dS·m⁻¹.

The comparison of the error distribution at different EC_p ranges showed that Rhoades and Archie did not give significantly different performances. Nevertheless, the Rhoades model showed a larger RMSE at high EC_p values than at low ones (EC_p < 10 dS·m⁻¹: RMSE = 0.78 dS·m⁻¹; EC_p > 10 dS·m⁻¹: RMSE = 1.17 dS·m⁻¹). On the other hand, the Archie model showed a greater consistency over the two salinity ranges (EC_p < 10 dS·m⁻¹: RMSE = 0.69 dS·m⁻¹; EC_p > 10 dS·m⁻¹: RMSE = 0.54 dS·m⁻¹).

As reported above, the “general” formulations of Rhoades and Archie showed overall similar performances. As already stated experimental θ values were used in the two equations. A simultaneous calibration was then done estimating EC_p and θ from EC_a and ε_r readings by substituting the “general” logarithmic θ model (Equation (15)) within Rhoades and Archie “general” models. The W₁ weight (Equation (12)) was set to 0.5, thus improving the EC_p estimation without notably worsening the θ evaluation.

The combined logarithmic θ model and Rhoades reads: (22) E C p = E C a ( a ′ Rhoades + a ″ Rhoades × E C a ) × ( 1 − 0.742 × ln ( ε r ) ) × π Rhoadeswith: (23) a ′ Rhoades = − 0.427 − 4.0 ⋅ 10 − 5 × S O C ( % ) (24) a ″ Rhoades = 0.024 − 0.008 × clay ( % ) sand ( % ) (25) π Rhoades = ( 0.074 × CaCO 3 ( % ) − 1.354 ) + ( − 0.132 × CaCO 3 ( % ) + 3.232 ) × ϕwhere a′_Rhoades, a″_Rhoades, and π_Rhoades are the fitting parameters defined in Equations (16), (17), and (19) during the independent calibration of θ and EC_p.

Similarly, the combined logarithmic θ model and Archie becomes: (26) E C p = 0.466 × E C a ϕ m Archie ( ( a ′ Archie + a ″ Archie × E C a ) × ( 1 − 0.738 × ln ( ε r ) ) ϕ ) n Archiewith: (27) a ′ Archie = − 0.427 − 0.006 × S O C ( % ) (28) a ″ Archie = 0.036 − 0.012 × clay ( % ) sand ( % ) (29) m Archie = − 2.8 ⋅ 10 − 4 × S O C ( % ) + 4.134 × S O C ( % ) S O C ( % ) + 0.719 (30) n = − 0.697 + 0.040 × sand ( % )where a′_Archie, a″_Archie, m_Archie, and n_Archie are the fitting parameters originally defined in Equations (16), (17), (21), and (20).

The calibration of Equation (22) yielded RSME values for θ and EC_p of 0.048 m³·m⁻³ and 0.77 dS·m⁻¹, respectively. Better overall results were obtained by Equation (26): RMSE = 0.046 m³·m⁻³ and RMSE = 0.63 dS·m⁻¹ for θ and EC_p, respectively. It is worth noting that the simultaneously calibrated parameters were very close to the independently calibrated ones.

A bootstrap validation was done on the simultaneous calibrations. A total of 5,000 iterations were operated for both Equations (22) and (26). Table 4 shows the variations of the slope and intercept of the fitting linear regression between observed and predicted values. Soil water content was correctly predicted by both the equations: the slope and intercept medians of the observed-estimated relationships were fairly close to 1 and 0, respectively. EC_p predictions were less accurate, generally overestimated by Equation (22) and underestimated by Equation (26) (Table 4).

According to the Kolmogorov-Smirnov test, significant differences were observed between the two equations. The Archie-based model provided significantly lower RMSE values on the validation sets for both θ (p < 0.01) and EC_p (p < 0.01) (Figure 5(a,b)). Equations (22) or (26) provided similar maximum errors for water content, with maximum RMSE of 0.08 m³·m⁻³ and 0.09 m³·m⁻³, respectively. On the other hand, Equation (22) produced a maximum EC_p error higher than that of Equation (26) (451.42 dS·m⁻¹vs. 211.26 dS·m⁻¹). The overall more accurate prediction of the system implementing Archie can be justified by the more flexible functional form of the EC_p model allowed by the two exponential parameters.