# Towards Improved Field Application of Using Distributed Temperature Sensing for Soil Moisture Estimation: A Laboratory Experiment

^{*}

## Abstract

**:**

## 1. Introduction

_{s}by a soil-specific transfer model predicting the soil thermal conductivity λ(ϴ) from the volumetric water content [17,18]. The dual probe heat pulse (DPHP) approach can also be adapted to fiber optic measurement by burying a heater and a sensing cable in the soil at a fixed distance. Benítez-Buelga et al. [19] showed the feasibility of the dual probe heating approach for estimating soil thermal properties using the infinite line source approach. A short heat pulse propagates from the heater cable against the sensing cable and produces a time shifted and temperature attenuated signal, depending on the thermal properties of the materials in between, and allows the measurement of λ

_{s}and the soil heat capacity C

_{s}.

_{s}, the water content can be derived from the soil heat capacity C

_{s}[20]. Secondly, a problem arising with heated fiber optic cables is the relatively large ratio of cable radii to cable spacing [19]. As a result, the cable influences itself during heat propagation, as it consists of multiple layers with varying thermal properties, and the measured heat capacity is a mixture of the heat capacity of all layers. The influence of the cable can be considered using the areal fraction of the geometry as weighted proportion to calculate the overall heat capacity of the cable. Thirdly, the standard approach for deriving thermal properties of heat pulses is the usage of the infinite line source analytic solution of the heat equation, which assumes a zero radius. However, soil DTS fiber optic cable has a relatively thick cable diameter with high crush resistance and rodent protection. Another problem might be that both single probe heat pulse and dual probe heat pulse DTS measurements suffer from poor soil-cable contact. For some sensors, the usage of porous blocks represents one possibility to eliminate this problem [21]. These problems are normally overcome by introducing an apparent spacing L

_{app}to take into account the probe geometry, materials, non-zero radius, and contact resistance [22]. Gathering L

_{app}requires another calibration process under known (dry or saturated) conditions, which limits the application in the field.

## 2. Materials and Methods

#### 2.1. Experimental Design

^{3}m

^{−3}.

^{3}m

^{−3}was covered. The measurements started in September 2017 and lasted until the middle of November 2017 with a total of 14 measurement days. Despite the ambition to eliminate any sources of uncertainty in the experimental design, uncertainties remained. First, there were uncertainties in the positioning of the cable, which might have changed during the filling and shrinking process of the soil. The shrinking process also provoked changes of the soil bulk density. Therefore, bulk density measurements were repeated after the soil was dried up completely again. Calculations were done twice, first using both bulk densities independently, and then comparing the results. Another source of uncertainty was the changing ambient air temperature, which influenced soil temperature heat fluxes but represented real conditions for potential real time applications. As the sources of uncertainties were known and also appear in the field, they must be considered in the interpretation of the results.

#### 2.2. Determination of Soil Properties

#### 2.2.1. Thermal Properties

_{0}is the heating time, q is the dissipated power per unit length (W/m), κ is the thermal diffusivity (m

^{2}/s), and Ei is the exponential integral function. However, this solution assumes a zero radius of the heater, which is practically impossible and normally considered through an apparent spacing L

_{app}, calibrated under known condition. To avoid this calibration procedure, we used the reformulated heat equation for an infinite line source with finite probe radii, finite heat capacity, and infinite thermal conductivity [22,23] to calculate the thermal properties. Contact resistance between cable and soil might influence the outcome but is not considered in this calculation. The analytic solution in the Laplace transform domain can be written as

_{2}is the temperature at the sensor, p is the transform variable of the Laplace transform, v

_{f}is the transfer function, Γ

_{1}is the heating function, K

_{0}is the modified Bessel function of the second kind of order 0, L is the distance between the two centers of the cable, λ the thermal conductivity (W m

^{−1}K

^{−1}), and

_{f}is formulated as

_{1}is the modified Bessel function of the second kind of order 1, r

_{n}is the cable radius of the heating cable (n = 1) and the sense cable (n = 2), and b

_{n}is the ratio of the volumetric heat capacity of the cable C

_{n}divided by the volumetric heat capacity of the soil C. The heating function Γ of an instantaneous heating impulse of certain duration t

_{0}is in the Laplace domain defined as

^{−1}). The analytic solution of the heat equation in the Laplace domain was evaluated using MATLAB (version 8.5.0). The inversion procedure from the Laplace domain to the time domain was done by using the Stehfest algorithm [27] with 16 coefficients, as recommended in [23].

_{n}was calculated by splitting the cable into its components and assigning them specific heat capacities (Table 1). In order to get the volumetric heat capacities, the subcomponents were weighed with a precision balance, and its volume was calculated from the cable geometry. The heat capacity of the cable is defined as the weighted sum of the volumetric fractions and the volumetric heat capacities of the cable components [19].

_{1}is defined to test the influence of different heating scenarios on the outcomes. They differ in heating time t

_{0}and power q and its range aligned to heating strategy experiences from former experiments [14,15,28]. Table 2 shows the different heating scenarios used in this experiment.

_{2}as a function of the of the distance and time and the soil thermal parameters C and λ. To avoid time consuming optimization procedure of C and λ, a look up table of resulting temperature functions Γ

_{2}(t) at the sensor cables were calculated within a range of potential values of C

_{s}and λ. For C

_{s}a range of 1 × 10

^{6}–5 × 10

^{6}Jm

^{−3}K

^{−1}with 10

^{5}Jm

^{−3}K

^{−1}steps was considered. Combined with λ ranging from 0 to 1.5 Wm

^{−1}K

^{−1}with 0.05 Wm

^{−1}K

^{−1}steps a total set A of 1271 temperature functions Γ

_{2}at distance L result for every heating scenario Γ

_{1}was used. These temperature functions were compared to the measured temperature signal from the DTS device.

#### 2.2.2. Measured vs. Modeled Data

_{maxM}at a certain time t

_{maxM}after heating has started at the sensor cable [2,19]. These values were compared to the simulated values (T

_{maxS}, t

_{maxS}) of the calculated response functions Γ

_{2}in two steps. According to the law of conservation of energy and the static geometry of the system, T

_{max}only varies by variations of C

_{s}. In a first step,

_{2}was conducted. The modeled values of the heating functions Γ

_{2}were compared to the measured values by a best fit function using the root mean squared error. Different evaluation times t

_{eval}were tested considering only the heating phase at the sensing cable, parts of the heating phase, or the heating and cooling phase for the regression approach (Figure 2).

#### 2.2.3. Volumetric Water Content

^{−3}), c

_{s}is the soil solid specific heat capacity (J kg

^{−1}K

^{−1}), ρ

_{w}is the density of water (kg m

^{−3}), c

_{w}is the specific heat capacity of water (J kg

^{−1}K

^{−1}), and ϴ the volumetric water content (m

^{3}m

^{−3}). Knowing the soil bulk density, the soil volumetric heat capacity, and the soils’ specific heat capacity, the volumetric water content can be determined by reformulating Equation (8):

_{app}[19]. The determination of the volumetric water content strongly depends on the accurate measurement of the heat capacity of the soil. Choosing an infinite or finite line source solution as well as using the amplitude values T

_{max}and t

_{max}or a regression approach for the determination of C might be decisive in the quality of the results of ϴ.

## 3. Results

#### 3.1. Signal Interpretation

_{maxM}and its corresponding time after starting the measurement t

_{max}was influenced by this uncertainty, leading to a small mismatch of the real temperature peak by overlapping of the temperature with the signal noise (Figure 2). To reduce the influence of measurement errors, noise reduction methods can be applied to the raw temperature signal. Simple smoothing with a simple moving average was considered inappropriate and would cause a flattening of the heating curve and the peak T

_{max}, as well as a possible shift of t

_{max}. Therefore, a Savitzky–Golay (SG) smoothing filter was applied to eliminate positive outliers without provoking an excessive smoothing of the temperature signal. The SG filter was specified with a polynomial function of order 3 with a frame length of seven measurements. The usage of the Savitzky–Golay filter showed satisfactory elimination of signal noise. In Figure 2, an example of the effect on the peak detection is shown, comparing the response function Γ

_{2}of a 40 Watt, 150 s heat impulse without raw signal treatment (blue) and the filtered signal (red) with the modeled response function. The maximum value of the raw measurement (blue dot) was influenced by a positive outlier, increasing T

_{max}and shifting it. The smoothed Savitzky–Golay filtered signal (red) coincided better with the reference amplitude of the modeled solution (yellow dot) optimized for the heating phase.

_{2}.

#### 3.2. Volumetric Water Content

_{s}, and the calculated heat capacity of the soil, and is calculated as outlined in Equation (9). The measured soil bulk density ranged from 1.45 Mg m

^{−3}under dry condition before wetting to 1.6 Mg m

^{−3}of the compacted, parched, dry soil. For c

_{s}, a value of 801 J kg

^{−1}K

^{−1}was estimated from literature values based on the results of the soil texture analysis. The volumetric heat capacity C was calculated as explained in Section 2.2.1. The outcomes of the volumetric water content with different bulk density and different calculation of the thermal properties were compared to the FDR sensor measurements and are shown in Figure 4. As the validation measurements refer to the volumetric water content, these results are presented first. In Section 3.3 the underlying soil properties of the best-fit solution for the FDR measurements are presented.

_{DTS}without calibration, independently of using the T

_{maxM}and t

_{maxM}approach (Figure 4a) or using a regression approach (Figure 4b). Figure 4a additionally shows the sensitivity of using different bulk densities, resulting in a shift of the volumetric water content. The red points show ϴ

_{DTS}with an initially measured bulk density of 1.45 Mg m

^{−3}and the blue points for a bulk density of 1.6 Mg m

^{−3}. Independently of ρd, both results had a high correlation coefficient, r = 0.84 (Table 3), but overestimated the water content. Differences were shown in the mean bias, which improved when a higher bulk density ρd was used. However, at higher water content, a drop of the calculated ϴ

_{DTS}was recognized with better results from ρd = 1.45 Mg m

^{−3}. This seems logical, because of the experimental design, as the dry soil before saturation was more loosely bedded than after drying, provoked by a shrinking process of the soil during drying up. This effect was also be observed by the appearance of shrinkage cracks at a water content ϴ = 0.25 m

^{3}m

^{−3}and the effect was also verified by the bulk density measurements before saturation (ρd = 1.45 kg dm

^{−3}) and after drying up (ρd = 1.6 kg dm

^{−3}). Therefore, it is recommended to use the initial bulk density for high water content and the bulk density after drying up for lower water content.

^{−3}with a considered t

_{eval}of 600 s and 1000 s. t

_{max}fluctuated between 400 and 600 s. Thus, the first t

_{eval}mainly included the heating phase and the second one the heating and cooling phases. Time spans between 400 and 1500 s were tested, and the best results were obtained for 600 s.

#### 3.3. Thermal Properties

_{eval}= 600 s. In Figure 5, the means of the volumetric heat capacity C and the thermal conductivity λ at a given volumetric water content with their respective uncertainties are shown. The coefficient of variations ranged from 3% to 17%. The results were consistent with literature values for the same soil types at given water content [30,31].

^{3}m

^{−3}. The latter might be explained by measurement uncertainties themselves. The support of the DTS measurement represented a 0.5 m path of the cable, although the FDR measurement supported only a single point. Additionally, the values represented the mean at a given water content, rising from different sensors, differing slightly in contact resistance, spacing, and accuracy. The FDR medium calibration accuracy was determined by ±0.02 m

^{3}m

^{−3}. Given this value, it was possible to reconfigure the results considering the accuracy of the FDR sensors (Figure 6). C and λ showed a strong correlation. Uncertainties showed slight increases at higher water contents. This was caused by the lower sensitivity of the temperature curve compared to changes at the upper range of C (Figure 7).

#### 3.4. Comparison to Infinite Line Source Solution

## 4. Discussion and Conclusions

_{s}is required a priori to get an estimate of the volumetric water content.

_{max}and t

_{max}regression approach. For the regression approach, it is important to include the heating phase and the peak in the time span. It is not recommended to include the cooling phase, where longitudinal thermal conduction might influence the measurements. This effect, however, is likely to be irrelevant at the field-scale, where the fiber optic cable can be implemented further outside the region of interest. Filtering of the noisy data is recommended, especially when using T

_{max}and t

_{max}for estimating ϴ. Regarding the heating impulse itself, a short but strong impulse is preferable. This produced a sharper temperature curve, which made the detection of T

_{max}and t

_{max}easier. The upper limit of the applied power must be chosen carefully to avoid a forced heating convection process. The estimation of ϴ was better at lower water contents, which arose from the higher sensibility of the temperature response curve at lower water contents.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

b_{n} | Ratio of volumetric heat capacity of the cable with volumetric heat capacity of the soil (–) |

c | Specific heat capacity of the soil (J kg^{−1} K^{−1}) |

c_{s} | Soil solid specific heat capacity (J kg^{−1} K^{−1}) |

c_{w} | Specific heat capacity of water (J kg^{−1} K^{−1}) |

C_{s} | Volumetric heat capacity of the soil (J m^{−3} K^{−1}) |

C_{1} | Heat capacity of the heating cable (J m^{−3} K^{−1}) |

C_{2} | Heat capacity of the sensing cable (J m^{−3} K^{−1}) |

p | Transform variable of the Laplace transform |

v_{f} | Transfer function |

K_{0} | Modified Bessel function of the second kind of order 0 |

K_{1} | Modified Bessel function of the second kind of order 1 |

L | Distance between the two centers of the cables (m) |

L_{app} | Apparent spacing between the two centers of the cables (m) |

q | Power released per unit length cable (W m^{−1}) |

r_{1} | Cable radius of the heating cable (m) |

r_{2} | Cable radius of the sensing cable (m) |

t_{0} | Heating time (s) |

t_{eval} | Evaluation time for regression approach (s) |

T_{max} | Temperature amplitude at the sensor cable (K) |

t_{max} | Time of t_{max} after heating has started (s) |

Γ_{1}(t) | Heating function at the heater cable |

Γ_{2}(t) | Heating function at the sensor cable |

ϴ | Volumetric water content (m^{3} m^{−3}) |

κ | Thermal diffusivity (m^{2} s^{−1}) |

λ_{s} | Thermal conductivity of the soil (W m^{−1} K^{−1}) |

ρd | Soil bulk density (kg m^{−3}) |

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**Figure 1.**(

**a**) Schematic cross section profile of cable positioning; red: heating cable, grey: sensing cables; cables are separated by wooden spacers, free drainage at the bottom to avoid impounding; (

**b**) photo of the 5 m × 0.3 m × 0.3 m perforated wooden box while burying the cables. Heating cable in black, two horizontal distanced sensor cables in white. On the right the frequency-domain-response (FDR) validation sensors are buried.

**Figure 2.**Raw signal at the sensing cable from the distributed temperate sensing (DTS) measurements with a 40 Watt heat pulse of 150 s duration in blue vs. filtered raw signals with a Savitzky–Golay Filter (3rd polynomial, seven frames) in red vs. optimum least RMSE semi-analytic solution in yellow. Points show the resulting maximum, respectively.

**Figure 4.**Comparison of calculated volumetric water content ϴ from DTS (shaded = 2 × standard deviation) measurements resulting from the finite semi-analytic solution with measurements from FDR sensor measurements for heating scenario 40 Watt and 150 s: (

**a**) results extracting T

_{max}and t

_{max}for a bulk density of 1.45 Mg m

^{−3}(red) and 1.6 Mg m

^{−3}(blue) and (

**b**) results for regression curve for t

_{eval}= 1000 s (red) and t

_{eval}= 600 s (blue).

**Figure 5.**Volumetric heat capacity C (

**a**) and thermal conductivity λ (

**b**) vs. volumetric water content ϴ from the FDR sensors for a 40 Watt heat pulse of 150 s extracted from a correlation–regression approach (shaded = 2 × standard deviation of measurement uncertainties) for a 600 s time span.

**Figure 6.**Volumetric heat capacity C (

**a**) and thermal conductivity λ (

**b**) vs. volumetric water content ϴ from the FDR sensors for a 40 Watt heat pulse of 150 s extracted from a correlation–regression approach for a 600 s time span, averaged over the medium-specific calibration accuracy of the FDR sensors of ±0.02 m

^{3}m

^{−3}.

**Figure 7.**Calculated response signal with the semi-analytic solution for a 40 Watt heat pulse of 150 s for different volumetric heat capacities (1 × 10

^{6}to 6 × 10

^{6}Jm

^{−3}K).

**Figure 8.**Comparison of calculated volumetric water content ϴ from DTS (shaded = 2 × standard deviation) measurements resulting from infinite line source solution without calibration (blue) and with calibration of the apparent spacing L

_{app}(red).

Component | Material | Specific Heat Capacity c (J kg^{−1} K^{−1}) | Volumetric Heat Capacity (J m^{−3} K^{−1}) |
---|---|---|---|

HEATER CABLE | |||

Coating Heater cable | PUR ^{1} | 1760 | 2.3 × 10^{6} |

Aramid | Aramid | 1200 | 2.1 × 10^{2} |

Copper Coating | PE ^{2} | 674 | 3.6 × 10^{6} |

SENSE CABLE | |||

Coating Sense cable | PUR | 1760 | 2.3 × 10^{6} |

Aramid | Aramid | 1200 | 2.1 × 10^{2} |

Fiber coating | Metal | 510 | 3.9 × 10^{6} |

^{1}Polyurethane,

^{2}Polyethylene.

Scenario ID | Power q (W m^{−1}) | Heating Time t_{0} (s) | Total Dissipated Energy (J) |
---|---|---|---|

1 | 10 | 300 | 3000 |

2 | 10 | 600 | 6000 |

3 | 20 | 300 | 6000 |

4 | 20 | 150 | 3000 |

5 | 20 | 100 | 2000 |

6 | 40 | 100 | 4000 |

7 | 40 | 150 | 6000 |

**Table 3.**Correlation coefficient and mean bias of selected representative measurements of ϴ

_{DTS}and ϴ

_{FDR.}

Finite Radius Line Source Solution | Infinite | ||||
---|---|---|---|---|---|

T_{max} t_{max} | Regression | r_app | |||

ρd = 1.45 | ρd = 1.6 | t_{eval} = 600 s | t_{eval} = 1000 s | ||

CorrCoeff | 0.85 | 0.85 | 0.88 | 0.8 | 0.82 |

MB | 0.08 | 0.06 | 0.04 | 0.13 | 0.04 |

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**MDPI and ACS Style**

Apperl, B.; Bernhardt, M.; Schulz, K.
Towards Improved Field Application of Using Distributed Temperature Sensing for Soil Moisture Estimation: A Laboratory Experiment. *Sensors* **2020**, *20*, 29.
https://doi.org/10.3390/s20010029

**AMA Style**

Apperl B, Bernhardt M, Schulz K.
Towards Improved Field Application of Using Distributed Temperature Sensing for Soil Moisture Estimation: A Laboratory Experiment. *Sensors*. 2020; 20(1):29.
https://doi.org/10.3390/s20010029

**Chicago/Turabian Style**

Apperl, Benjamin, Matthias Bernhardt, and Karsten Schulz.
2020. "Towards Improved Field Application of Using Distributed Temperature Sensing for Soil Moisture Estimation: A Laboratory Experiment" *Sensors* 20, no. 1: 29.
https://doi.org/10.3390/s20010029