# Derivation of Heat Conductivity from Temperature and Heat Flux Measurements in Soil

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

- ${C}_{1}>0,\phantom{\rule{3.33333pt}{0ex}}{C}_{2}=0$; the data of three temperature loggers at different depths are used ($\mathsf{\Phi}\left({a}^{\prime}\right)$ is RMSE of temperature squared) (this method coincides with that used in [25,26,27]), hereafter referred to as TEMP method (hereafter, the depth of measurements for this setting is denoted as ${z}_{3}={z}_{m}$);
- ${C}_{1}=0,\phantom{\rule{3.33333pt}{0ex}}{C}_{2}>0$; the data of two temperature sensors and a heat flux plate located in between are used ($\mathsf{\Phi}\left({a}^{\prime}\right)$ is the RMSE of heat flux squared), hereafter referred to as the FLUX method (hereafter, the depth of measurements for this setting is denoted as ${z}_{4}={z}_{m}$).

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Errors of Inverse Solution according to Fourier Law

## References

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1. | According to Fourier solution, a is a function of a ratio of temperature magnitudes at two depths; systematic temperature shifts do not change magnitudes and hence a. |

**Figure 1.**The landscape view and setup of measurements on Mukhrino bog (Khanty–Mansiysk region, Russia) in summer, 2019. The red rectangle shows the vertical string of temperature and heat flux sensors in the moss layer.

**Figure 2.**Moss temperature series measured at top (5 cm), bottom (25 cm) and middle (15 cm), of the test moss layer, and simulated in the middle of the layer at the optimal value of the temperature diffusivity coefficient ${a}_{0}$.

**Table 1.**Inverse problem solution as a function of input data errors. $\delta $(·) is a mean error of variable (·). True solution is $a\rho c={a}_{w}\rho c=0.561$ Wm${}^{-1}$K${}^{-1}$. The solutions with zero input data errors and with maximal $|{a}_{0}-{a}_{w}|$ are shown.

Solution under different temperature mean errors (TEMP setup) | ||||

$\delta {T}_{1}$, K | $\delta {T}_{m}$, K | $\delta {T}_{2}$, K | ${a}_{0}\rho c$, Wm${}^{-1}$ K${}^{-1}$ (relative error) | RMSE, ${}^{\xb0}$C |

0 | 0 | 0 | 0.559 (−0.4%) | 0.10 |

−0.1 | 0.1 | −0.1 | 0.570 (+1.6%) | 0.23 |

−0.1 | 0.1 | 0 | 0.567 (+1.1%) | 0.18 |

0 | −0.1 | 0.1 | 0.553 (−1.4%) | 0.18 |

0 | 0.1 | −0.1 | 0.571 (+1.8%) | 0.18 |

0.1 | −0.1 | −0.1 | 0.558 (−0.5%) | 0.15 |

0.1 | −0.1 | 0 | 0.546 (−2.7%) | 0.18 |

0.1 | −0.1 | 0.1 | 0.546 (−2.7%) | 0.22 |

Solution under different temperature sensor depth errors (TEMP setup) | ||||

$\delta {z}_{1}$, cm | $\delta {z}_{m}$, cm | $\delta {z}_{2}$, cm | ${a}_{0}\rho c$, Wm${}^{-1}$ K${}^{-1}$ (relative error) | RMSE, ${}^{\xb0}$C |

0 | 0 | 0 | 0.564 (+0.5%) | 0.10 |

−1 | +1 | −1 | 0.787 (+40.3%) | 0.31 |

−1 | +1 | 0 | 0.801 (+42.8%) | 0.23 |

0 | −1 | +1 | 0.441 (−21.4%) | 0.23 |

0 | +1 | −1 | 0.655 (+16.8%) | 0.24 |

+1 | −1 | 0 | 0.366 (−34.8%) | 0.23 |

+1 | −1 | +1 | 0.357 (−36.3%) | 0.30 |

Solution under different temperature and heat flux mean errors (FLUX setup) | ||||

$\delta {T}_{1}$, K | $\delta {F}_{m}$, % | $\delta {T}_{2}$, K | ${a}_{0}\rho c$, Wm${}^{-1}$ K${}^{-1}$ (relative error) | RMSE, W m${}^{-2}$ |

0 | 0 | 0 | 0.563 (+0.4%) | 0.99 |

−0.1 | +15 | 0 | 0.654 (+16.6%) | 1.11 |

−0.1 | +15 | +0.1 | 0.671 (+19.6%) | 1.16 |

0 | −15 | −0.1 | 0.467 (−16.8%) | 1.06 |

0 | +15 | +0.1 | 0.654 (+16.6%) | 1.11 |

+0.1 | −15 | −0.1 | 0.456 (−18.7%) | 1.13 |

+0.1 | +15 | +0.1 | 0.638 (+13.7%) | 1.08 |

Solution under different heat flux plate depth errors (FLUX setup) | ||||

$\delta {z}_{1}$, cm | $\delta {z}_{m}$ cm | $\delta {z}_{2}$, cm | ${a}_{0}\rho c$, Wm${}^{-1}$ K${}^{-1}$ (relative error) | RMSE, W m${}^{-2}$ |

0 | 0 | 0 | 0.560 (−0.2%) | 1.01 |

−1 | 0 | −1 | 0.574 (+2.3%) | 1.16 |

−1 | +1 | −1 | 0.586 (+4.5%) | 1.38 |

−1 | +1 | 0 | 0.616 (+9.8%) | 1.34 |

−1 | +1 | +1 | 0.641 (+14.3%) | 1.22 |

0 | −1 | +1 | 0.560 (−0.2%) | 1.26 |

+1 | −1 | −1 | 0.481 (−14.3%) | 1.31 |

+1 | −1 | 0 | 0.499 (−11.1%) | 1.41 |

+1 | −1 | +1 | 0.513 (−8.6%) | 1.55 |

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**MDPI and ACS Style**

Stepanenko, V.; Repina, I.; Artamonov, A.
Derivation of Heat Conductivity from Temperature and Heat Flux Measurements in Soil. *Land* **2021**, *10*, 552.
https://doi.org/10.3390/land10060552

**AMA Style**

Stepanenko V, Repina I, Artamonov A.
Derivation of Heat Conductivity from Temperature and Heat Flux Measurements in Soil. *Land*. 2021; 10(6):552.
https://doi.org/10.3390/land10060552

**Chicago/Turabian Style**

Stepanenko, Victor, Irina Repina, and Arseniy Artamonov.
2021. "Derivation of Heat Conductivity from Temperature and Heat Flux Measurements in Soil" *Land* 10, no. 6: 552.
https://doi.org/10.3390/land10060552