# Estimating Pore Water Electrical Conductivity of Sandy Soil from Time Domain Reflectometry Records Using a Time-Varying Dynamic Linear Model

^{1}

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## Abstract

**:**

_{p}) from soil bulk electrical conductivity (σ

_{b}) in ecological and hydrological applications, a good method of doing so remains elusive. The Hilhorst concept offers a theoretical model describing a linear relationship between σ

_{b}, and relative dielectric permittivity (ε

_{b}) in moist soil. The reciprocal of pore water electrical conductivity (1/σ

_{p}) appears as a slope of the Hilhorst model and the ordinary least squares (OLS) of this linear relationship yields a single estimate ($\widehat{1/{\sigma}_{p}}$) of the regression parameter vector (σ

_{p}) for the entire data. This study was carried out on a sandy soil under laboratory conditions. We used a time-varying dynamic linear model (DLM) and the Kalman filter (Kf) to estimate the evolution of σ

_{p}over time. A time series of the relative dielectric permittivity (ε

_{b}) and σ

_{b}of the soil were measured using time domain reflectometry (TDR) at different depths in a soil column to transform the deterministic Hilhorst model into a stochastic model and evaluate the linear relationship between ε

_{b}and σ

_{b}in order to capture deterministic changes to (1/σ

_{p}). Applying the Hilhorst model, strong positive autocorrelations between the residuals could be found. By using and modifying them to DLM, the observed and modeled data of ε

_{b}obtain a much better match and the estimated evolution of σ

_{p}converged to its true value. Moreover, the offset of this linear relation varies for each soil depth.

## 1. Introduction

_{p}) as an indicator of the soil salinity; however, they are labour- and cost intensive. There is no evidence that all ions are collected in the sample extract [3]. For soil salinity assessment, it is important to look for practical methods that evaluate the soil salinity state temporally and spatially. These methods help to correctly evaluate soil salinity evolution and reasonably predict its values [4,5,6,7,8,9]. In recent times, soil electromagnetic sensors have been used to estimate bulk electrical conductivity (σ

_{b}). Then, methods are required to transform σ

_{b}to σ

_{p}[3,6,10].

_{b}of a soil: (i) solid phase pathway through soil particles that are continuous contact with one another, (ii) liquid phase pathway through dissolved ions in the soil water inhabiting the large pores, and (iii) a liquid–solid interphase pathway through exchangeable cations like surfaces of clay minerals. Electrical conductivity (EC) in the liquid phase (σ

_{p}) is used to estimate the soil salinity, a high EC refers to a high concentration of soluble salts, and vice versa. The σ

_{p}could be estimated if the relationship between σ

_{p}, σ

_{b}, and water content (θ) is fixed [12,13,14]. The discovered linear correlation between the soil relative dielectric permittivity (ε

_{b}) and σ

_{b}values [15] enabled Hilhorst [3] to convert σ

_{b}to σ

_{p}by using a theoretical model. According to Hilhorst, σ

_{p}can be determined from the equation:

_{p}is the pore water electrical conductivity (dS/m); ε

_{p}is the relative dielectric permittivity of the soil pore water (dimensionless), ε

_{b}is the relative dielectric permittivity of the bulk soil (dimensionless, relative dielectric permittivity is dimensionless since it is a ratio of permittivity of medium to the permittivity of free space), σ

_{b}is the bulk electrical conductivity (dS/m), ε

_{σb=0}is the relative dielectric permittivity of the soil when the bulk electrical conductivity is 0 (dimensionless). However, ε

_{σb=0}appears as an offset of the linear relationship between ε

_{b}and σ

_{b}. The Hilhorst model [3] concluded that his method could be validated for water contents between 0.10 and saturation and for a conductivity of the pore water up to 0.3 S m

^{−1}. He found that ε

_{σb=0}depends on soil type and varies between 1.9 and 7.6. He recommended using 4.1 as a generic offset. Many studies applied the deterministic Hilhorst model [3] in their experiments to convert σ

_{b}into σ

_{p}but they did not use the same offset to achieve their study objective. For example, some studies concluded their work by using different offsets (within the range of 3.67 to 6.38) according to the soil type [10]. The producer of capacitance soil moisture sensors 5TE [16] recommends the use of an offset ε

_{σb=0}of 6 while another study found that an offset ε

_{σb=0}= 6 does not present a good linear relationship between ε

_{b}and σ

_{b}[17]. The WET sensor (Delta-T Device Ltd., Cambridge, UK) is a frequency domain dielectric sensor. It has been designed to estimate the σ

_{p}based on the Hilhorst model [3] and incorporate the standard offset ε

_{σb=0}= 4.1 of the model in the software of the device. By applying the Hilhorst model [3] in a saline gypsum-influenced soil, the accuracy of the WET sensor in predicting σ

_{p}was very poor when using the offset model =4.1 [18]. Another study used a WET sensor for experimental measurements in the laboratory using four different soils (sand, sandy loam, loam, and clay) [9] and found that the offset depends on both soil type and σ

_{p}, where it becomes larger for larger σ

_{p}. Moreover, oscillator frequency and sensor circuitry could affect the estimation of ε

_{b}and water content ($\theta $) [19].

- (i)
- many effects are left unknown since the objective of the model is to represent the main modes of system response,
- (ii)
- deterministic models are driven not by only our own control inputs but also disturbances which we can neither control nor model deterministically, and
- (iii)
- sensors do not offer exact readings of chosen quantities but present their own system dynamics and distortions as well and these devices are noise corrupted [20]. Despite the importance of computing σ
_{p}from σ_{b}, a good method for doing so remains elusive (Campbell [16], personal communication).

_{p}[22]. We used ε

_{b}and σ

_{b}observations to modify the Hilhorst deterministic model [3] to a stochastic model using a time-varying dynamic linear model and Klaman filter before studying the linear relationship between them.

_{b}and σ

_{b}in laboratory conditions where the soil is homogeneous. Then, we tried to use the Hilhorst model [3] to convert σ

_{b}to σ

_{p}. Later, we could show the weakness of applying the deterministic Hilhorst model [3] even in homogeneous soils. Thus, we are aiming to adapt this approach to a stochastic model under laboratory conditions. Thus, we used one homogeneous soil type to accurately estimate the changes in σ

_{p}over time and to conclude whether the model offset is constant or if it changes in one soil profile.

## 2. Material and Methods

#### 2.1. The Column Experiment

^{3}. The substrate was sand, 80% of which was fine sand. The water content during packing was approximately 4 m

^{3}/m

^{3}. The TDR and soil temperatures sensors were installed in four depths: 7, 21, 35, and 48 cm. Since the soil is sand, the soil relative dielectric permittivity (ε

_{b}), bulk electrical conductivity (σ

_{b}), and temperature were measured every 5 min to obtain enough observations for modeling.

_{p}):

_{b}, σ

_{b}, and ε

_{p}are required to estimate σ

_{p}. Therefore, we used five irrigation events with two levels of KCL solution to obtain the variation of these variables over time for each depth. In total, 289 observations were made of σ

_{b}, ε

_{p}, and ε

_{b}for each soil depth and these were used to estimate both the offset ε

_{σb=0}of the modified Hilhorst model [3] and the evolution of σ

_{p}at its corresponding depth, of which 144 observations were used to validate their forecasts.

#### 2.2. Time-Varying Dynamic Linear Model

_{t}called the state process. The state process is assumed to be a Markov process, where past and future values of x

_{t}are independent, conditional on the present x

_{t}, ({x

_{s}, S > t}, and {x

_{s}, S< t} are independent on the x

_{t}), (ii) the observations, y

_{t}are independent given the states x

_{t}. This means that the dependence among the observations is generated by states. The dynamic linear model (DLM) or linear Gaussian state space model, in its simple form, employs a first-order, p-dimensional vector autoregression as the state equation:

_{t}directly, but only a linear transformed version of them with noise added, say:

_{t}is an m-dimensional vector, representing the observation at time t, ${A}_{t}$ is a q × p measurement or observation matrix. Equation (4) is called the observation equation, in which ${v}_{t}$, ${w}_{t}$ are the Gaussian white-noise errors. The evolution variances are ${V}_{t}$, ${W}_{t}$ and can be estimated from available data using maximum likelihood or Bayesian techniques.

_{b}, ε

_{p}, σ

_{p}, ε

_{b}, and ε

_{σb = 0}. The σ

_{p}and ε

_{σb = 0}are unobserved and they need to be estimated by the state Equation (3) as x

_{t}, while σ

_{b}, ε

_{p}, and ε

_{b}are observed by the sensors (ε

_{p}is calculated from Equation (2) using soil temperature sensor data) and represented by observation Equation (4) as y

_{t}.

## 3. Results and Discussion

#### 3.1. Deterministic Model

_{σb = 0}) from this linear model after using measurements of ε

_{b}and σ

_{b}. For example, applying the ordinary least squares (OLS) on measurements of ε

_{b}and σ

_{b}obtained from soil column 2 data during the third irrigation at a depth of 21 cm, Table 1 shows that the offset of the linear relationship between ε

_{b}–σ

_{b}is 9.41. Further, the single estimate of the slope ($\widehat{1/{\sigma}_{p}}$) of the regression parameter vector (1/σ

_{p}) for the entire data set is very small. Thus, the estimated soil pore water electrical conductivity (σ

_{p}) is too high compared with the EC meter value, see Table 2. Afterward, we applied the Durbin–Watson test in order to test if there was any autocorrelation between the residuals of the regression. Table 3 shows that there is an extremely strong and positive autocorrelation, meaning that the result of that regression is not valid.

_{p}obtained from our modified model to the values of σ

_{p}obtained by the EC meter device, see Table 2.

#### 3.2. Time-Varying Linear Dynamic Model (LDM)

_{b}), bulk electrical conductivity (σ

_{b}), and the relative dielectric permittivity (ε

_{p}), while the unobservable data are the offset (ε

_{σb=0}) and pore water electrical conductivity (σ

_{p}). Equation (4) can be modified to the time-varying DLM as follows:

- The observation equation can be obtained by modifying the Hilhorst model [3] (written in Equation (5)) into a stochastic equation, in accordance with Equation (4) as follows:$${\left({\epsilon}_{b}\right)}_{t}={\left({\epsilon}_{{\sigma}_{b}}=0\right)}_{t}+{({\epsilon}_{p}\ast {\sigma}_{b})}_{t}{\left(\frac{1}{{\sigma}_{p}}\right)}_{t}+{v}_{t}\phantom{\rule{0ex}{0ex}}{v}_{t}~\mathcal{N}\left(0,{\sigma}_{v}{}^{2}\right)$$
- The state equation (unobservable data) in Equation (3) is ε
_{σb = 0}, and the slope, 1/σ_{p}. They can be converted to the unobservable state equation of the time-varying DLM according to Equation (3). The unobservable state equation can be arranged as follows:$$\{\begin{array}{c}{({\epsilon}_{{\sigma}_{b}}=0)}_{t}={({\epsilon}_{{\sigma}_{b}}=0)}_{t-1}\\ {\left(\frac{1}{{\sigma}_{p}}\right)}_{t}={\left(\frac{1}{{\sigma}_{p}}\right)}_{t-1}+{w}_{t}{w}_{t}~N(0,{\left({\sigma}_{w}\right)}_{t}^{2})\end{array}$$

_{σb = 0}as a constant. The actual value is related only to its past value. The slope 1/σ

_{p}changes over time and its actual value is related to its past value plus the Gaussian white-noise errors (${w}_{t}$). We applied the equation in reverse order to estimate the state variables (ε

_{σb = 0}and σ

_{p}) at all time points from a complete series of the soil relative dielectric permittivity (ε

_{b}). This process is known as smoothing.

_{b}, ε

_{p}, and σ

_{b}data needed for the Hilhorst model [3] is shown in Figure 4. By applying the Equations (6) and (7) using DLM and the Kalman filter on the eight time-series data, we see in Figure 5 the observed and predicted time series of the soil relative dielectric permittivity (ε

_{b}). The predicted and observed values of ε

_{b}agree reasonably well. The mean absolute prediction error (MAPE) for the time series never exceeded 0.02.

_{b}) is valid, the estimation of the electrical conductivity of pore water (σ

_{p}) and the offset ε

_{σb = 0}, see Equation (7), are also valid because they are used in the prediction of the soil relative dielectric permittivity (ε

_{b}) and have converged to their true values. The evolution of σ

_{p}over time obtained by DLM is presented in Figure 6; it shows the importance of using DLM because it obtained all the changes of σ

_{p}over time and not a single value of σ

_{p}for the entire data set. Another interesting aspect is that Figure 6 shows the changes in the model offset for each irrigation event at each depth. This finding is very important since it shows that the offset does not depend on the soil type [10,14,15,16] nor on the soil type and salinity [17] when two columns with the same type soil are used, as in this study. Moreover, in Figure 6 we put the corresponding value of σ

_{p}measured by the EC meter device for each depth according the irrigation event and soil column number.

_{p}values obtained from our modified Hilhorst model, see Figure 6, with the single corresponding EC value obtained from porous suction cups and measured by the EC meter device, see Table 2, we found that they agree very well (R

^{2}= 72%).

_{p}in two homogenous soil columns; first, we observed that the offset value of the Hilhorst model does not depend on the soil type and σ

_{p}and it changes in the same soil profile. Secondly, we obtained the changes in the estimated σ

_{p}over time and not just a single value as a coefficient for the entire data set. Third, the estimated changes in σ

_{p}occur instantly and save time and labor costs.

## 4. Conclusions

_{b}-σ

_{b}linear relationship to homogeneous soil column data obtained from TDR sensors. We found an extremely strong positive autocorrelation between the residuals of the regression analysis. When residuals are correlated, the least squares method is not the most efficient model coefficient estimator. By modifying the regression by a time-varying dynamic linear model (DLM), the match between the observed and modeled data of ε

_{b}is significantly improved and the estimated evolution of σ

_{p}converges to its true value. Moreover, in this study, we used two homogeneous soil columns with the same condition to show that the offset of the Hilhorst model [3] is not constant, as suggested for all moist soil or, as others suggested, that it is soil-type-dependent [10,14,15,16] or soil-type- and salinity-dependent [17]. We repeated the experiment to show that the offset changes even in the same soil type and the same conditions. A dynamic linear model enables the capture of the offset changes and it shows the importance of calculating it simultaneously when estimating σ

_{p}using the Hilhorst model. The next promising step would be programming and inserting these models into the TDR software in order to estimate the soil pore water electrical conductivity (σ

_{p}) from senor records directly.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Three conductance pathways for the σ

_{b}measurements, inspired by Wyllie and Southwick [11].

**Figure 3.**Bulk electrical conductivity (σ

_{b}) in the two soil columns for two irrigation events (N°.3 and N° 4) at two depths (21 cm and 35 cm). Series peaks are related to time irrigation.

**Figure 4.**Known variables for the Hilhorst model (σ

_{b}, ε

_{b}, and ε

_{p}); data from soil column 2, depth 35 cm and irrigation event N°4.

**Figure 5.**Observed and predicted soil relative dielectric permittivities according to the soil column number, depth, and irrigation event.

**Figure 6.**Estimation of the unobservable data (ε

_{σb = 0}and σ

_{p}) by applying the time-varying dynamic linear model (DLM) and the Kalman filter on the data according to the soil column number (col.), depth (dep.), and irrigation event (irg. Event), the corresponding of σ

_{p}by EC meter device is given for each estimated σ

_{p}.

Estimate | Std. Error | t Value | Pr (>|t|) | |
---|---|---|---|---|

ε_{σb = 0} | 9.411 | 8.591 × 10^{‒3} | 1095.4 | <2 × 10^{‒16} *** |

1/σ_{p} | 6.963 × 10^{‒}^{4} | 4.461 × 10^{‒6} | 156.1 | <2 × 10^{‒16} *** |

**Table 2.**Electrical conductivity of the soil solution (dS/m) according to soil column number, irrigation event, and depth (cm); it is collected by porous suction cups and measured by an electrical conductivity (EC) meter device.

Soil Column 1 | Soil Column 2 | ||||||
---|---|---|---|---|---|---|---|

Irrigation Event 3 | Irrigation Event 4 | Irrigation Event 3 | Irrigation Event 4 | ||||

Depth: 21 cm | Depth: 35 cm | Depth: 21 cm | Depth: 35 cm | Depth: 21 cm | Depth: 35 cm | Depth: 21 cm | Depth: 35 cm |

15.96 | 18 | 21.89 | 22.35 | 18.61 | 14.97 | 25.67 | 22 |

Lag | Autocorrelation | D-W Statistic | p-Value |
---|---|---|---|

1 | 0.852 | 0.278 | 0 |

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**MDPI and ACS Style**

Aljoumani, B.; Sanchez-Espigares, J.A.; Wessolek, G. Estimating Pore Water Electrical Conductivity of Sandy Soil from Time Domain Reflectometry Records Using a Time-Varying Dynamic Linear Model. *Sensors* **2018**, *18*, 4403.
https://doi.org/10.3390/s18124403

**AMA Style**

Aljoumani B, Sanchez-Espigares JA, Wessolek G. Estimating Pore Water Electrical Conductivity of Sandy Soil from Time Domain Reflectometry Records Using a Time-Varying Dynamic Linear Model. *Sensors*. 2018; 18(12):4403.
https://doi.org/10.3390/s18124403

**Chicago/Turabian Style**

Aljoumani, Basem, Jose A. Sanchez-Espigares, and Gerd Wessolek. 2018. "Estimating Pore Water Electrical Conductivity of Sandy Soil from Time Domain Reflectometry Records Using a Time-Varying Dynamic Linear Model" *Sensors* 18, no. 12: 4403.
https://doi.org/10.3390/s18124403