# Optimal Temporal Filtering of the Cosmic-Ray Neutron Signal to Reduce Soil Moisture Uncertainty

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Site

#### 2.2. Data

#### Generating Synthetic Neutron Signal for Selected Sites

#### 2.3. Analysis

#### 2.3.1. Moving Average

#### 2.3.2. Savitzky–Golay Filter

#### 2.3.3. Median Filter

#### 2.3.4. Kalman Filter

#### 2.3.5. Error Measurement

## 3. Results

#### 3.1. Evaluation of Filters’ Performance at the Four Sites

#### 3.2. Optimal Filter and Window Length

#### 3.3. Uncertainty Propagation from CRNS Standard Correction

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Map showing geographical location of stations used in the study. The countries where stations are located are highlighted in red.

**Figure 2.**Graphical representation of synthetic neutron flux ${N}_{syn}$ (gray) generated for Finland (

**a**), Gorigo (

**b**), Rollesbrioch (

**c**) and Conde (

**d**). The black line represents the neutron flux ${N}_{True}$ only subject to the soil moisture change, while red is the uncorrected neutron flux ${N}_{uncorrected}$ subject to atmospheric factors and soil moisture.

**Figure 3.**Schematic flow for generating synthetic neutron flux. Key steps for analysis of two scenarios when filters are applied after (scenario A) or before (scenario B) standard correction of synthetic neutron signal.

**Figure 4.**Performance of Kalman filter at the different station based on the MBE (

**a**), standard deviation (

**b**), RMSE (

**c**) and correlation (

**d**).

**Figure 5.**Performance of MA (red), MF (blue) and SG filters as a function of window length. The MBE, standard deviation, RMSE and correlation are presented in columns for SMEAR II (

**a**–

**d**), Gorigo (

**e**–

**h**), Rollesbroich (

**i**–

**l**) and Conde (

**m**–

**p**) for the four stations (in rows), respectively.

**Figure 6.**Time series of synthetic neutron flux (gray dot) and filtered flux using the MF, MA, SG and KF filtering techniques at SMEAR II (

**a**), Gorigo (

**b**), Rollesbroich (

**c**) and Conde (

**d**) sites. The black line represented the original neutron flux.

**Figure 7.**Relative percentage difference (

**b**) and (

**d**) of filter response to shape changes in soil moisture at Gorigo (

**a**), Rollesbroich (

**c**) sites, respectively. The pink shaded region represents the interested event.

**Figure 8.**Performance of the optimal window size of the different filtering approaches at SMEAR II (

**a**), Gorigo (

**b**), Rollesbroich (

**c**) and Conde (

**d**). The original neutron flux is denoted by the black dot.

**Figure 9.**Time series of observed soil moisture (black line) at Gorigo and estimated soil water content from filtered synthetic neutron count based on two scenarios. Scenario A (

**a**): Filters applied after atmospheric correction. Scenario B (

**b**): Filters applied before atmospheric correction.

Station | Lon/Lat | Bulk Density (g/cm${}^{3}$) | Rigidity Cut-Off (GV) | Other Site Information |
---|---|---|---|---|

Gorigo | 0.82/10.93 | 1.54 | 14.68 | Highly degraded grassland |

Loamy sand soil | ||||

Rollesbroich | 6.30/50.63 | 1.09 | 3.27 | Managed grassland |

Silty clay loam | ||||

SMEAR II | 24.29/61.84 | 0.85 | 1.11 | Homogenous Scots pine trees |

Silty sand [34] | ||||

Conde | −3.22/37.91 | 1.37 | 8.33 | Evergreen trees and shrubs. |

Clayey loam [35] |

**Table 2.**Standard deviations of true neutron flux, the synthetic signal and the Kalman-filtered signal for the four sites.

Station | True Neutron | Synthetic Neutron | KF-Filtered Neutron |
---|---|---|---|

Gorigo | 43.05 | 52.07 | 40.30 |

Rollesbroich | 31.16 | 37.99 | 29.99 |

SMEAR II | 13.31 | 26.34 | 12.63 |

Conde | 29.24 | 39.74 | 29.20 |

Station | MA (h) | MF (h) | SG-3 (h) | SG-4 (h) |
---|---|---|---|---|

Gorigo | 30 | 36 | 78 | 84 |

Rollesbroich | 18 | 18 | 30 | 48 |

SMEAR II | 36 | 42 | 54 | 84 |

Conde | 18 | 12 | 30 | 36 |

Filter | Scenario A (cm${}^{3}$/cm${}^{3}$) | Scenario B (cm${}^{3}$/cm${}^{3}$) |
---|---|---|

KF | 0.006 | 0.008 |

MA (30 h) | 0.006 | 0.009 |

SG-3 (78 h) | 0.007 | 0.009 |

SG-4 (84 h) | 0.007 | 0.008 |

MA (24 h) | 0.007 | 0.009 |

MF (36 h) | 0.007 | 0.009 |

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**MDPI and ACS Style**

Davies, P.; Baatz, R.; Bogena, H.R.; Quansah, E.; Amekudzi, L.K.
Optimal Temporal Filtering of the Cosmic-Ray Neutron Signal to Reduce Soil Moisture Uncertainty. *Sensors* **2022**, *22*, 9143.
https://doi.org/10.3390/s22239143

**AMA Style**

Davies P, Baatz R, Bogena HR, Quansah E, Amekudzi LK.
Optimal Temporal Filtering of the Cosmic-Ray Neutron Signal to Reduce Soil Moisture Uncertainty. *Sensors*. 2022; 22(23):9143.
https://doi.org/10.3390/s22239143

**Chicago/Turabian Style**

Davies, Patrick, Roland Baatz, Heye Reemt Bogena, Emmanuel Quansah, and Leonard Kofitse Amekudzi.
2022. "Optimal Temporal Filtering of the Cosmic-Ray Neutron Signal to Reduce Soil Moisture Uncertainty" *Sensors* 22, no. 23: 9143.
https://doi.org/10.3390/s22239143