# Robust Kalman Filter Soil Moisture Inversion Model Using GPS SNR Data—A Dual-Band Data Fusion Approach

^{1}

^{2}

^{3}

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^{5}

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Conventional Linear Regression Soil Moisture Inversion Model

#### 2.2. Robust Regression Soil Moisture Inversion Model

#### 2.3. Robust Kalman Filter Soil Moisture Inversion Model

## 3. Results

#### 3.1. Experiment Campaign

#### 3.2. Comparison of Results of Single-Band Univariate Regression on GPS L1—Scenario One

#### 3.2.1. Using Amplitude Observables

_{A1}is the correlation coefficients of amplitude on L1, and R

_{A2}is the correlation coefficients of amplitude on L2. It is clear to see that there is a linear correlation between amplitude and in situ soil moisture. For this satellite the correlation on the L2 band is better than L1. After training and predicting, we obtained the inversion results of the amplitude observable on GPS L2. We show the soil moistures retrievals in Figure 5 of our time series (47 days) for satellite PRN1 using the amplitude observable as input of the inverting model.

^{−2}m

^{3}/m

^{3}to ~29*10

^{−2}m

^{3}/m

^{3}). As demonstrated in Figure 5, the results of the Robust Kalman Filter model show much better agreement to the increase of in situ soil moisture after day 10 than the other two models. The second increase (days 20 to 30) of the in situ soil moisture is clearly identified by the models (lower amplitudes, Figure 5). After day 35 the models differ more from real measurements. However, we can see clearly that the Robust Kalman Filter model improved the estimation results and is better than robust/classical regressions.

#### 3.2.2. Using Phase Observables

_{P1}is the correlation coefficients of amplitude on L1, R

_{P2}is the correlation coefficients of amplitude on L2. According to Figure 6, when the in situ soil moisture increased, the observed phase on L1 and L2 showed a trend of decrease and vice versa, and both of them have a negative correlation. The inversion results of soil moisture using GPS L1 phase observable are shown in Figure 7.

#### 3.3. Comparison between Dual-Band Data Fusion Classical Regression Model, Robust Regression Model and Robust Kalman Model—Scenario Two

#### 3.3.1. Using Amplitude

#### 3.3.2. Using Phase

^{−2}m

^{3}/m

^{3}to ~29*10

^{−2}m

^{3}/m

^{3}). For the two models (robust regression and conventional regression) in general this growth has a dramatic reduction on day 13. The second increase (days 20 to 30) as compared to scenario one of the in situ soil moisture is clearly identified by the models. The robust regression model has a lower difference in general regarding this growth, and after day 30 the models strongly decrease. However, we can see clearly that the Robust Kalman Filter model improved the estimation results better than the robust/classical regressions. The two linear regression models differ with in situ soil moisture from day 35, but the Robust Kalman Filter model improved the estimation results and agreed with the decreasing trend.

#### 3.4. Multivariate Variable Dual-Band Data Fusion—Scenario Three

^{−2}m

^{3}/m

^{3}to ~29*10

^{−2}m

^{3}/m

^{3}). For the Robust Kalman Filter model in general this growth is much faster and reaches a maximum intensity after 10 days. The second increase (days 20 to 30) of the in situ soil moisture is clearly identified by all the three models, and after day 35 models of conventional method and robust regression method differ from real measurements, but they are better than the other two scenarios. However, we can see clearly that the Robust Kalman Filter model improved the estimation results better than the robust/classical regressions, and it is also better than the previous scenarios.

## 4. Analysis and Discussion

^{3}/m

^{3}. In article [39], the authors performed a 15-month observation which covered an entire growing cycle by two antennas and developed an inversion model on GPS L2C and L5 SNR, achieving a precision of 0.035 m

^{3}/m

^{3}for the whole meadow growing cycle, and of 0.018 m

^{3}/m

^{3}after grass cutting.

^{3}/m

^{3}and 2% m

^{3}/m

^{3}for the Robust Kalman Filter model.

^{3}/m

^{3}and 2% m

^{3}/m

^{3}for the Robust Kalman Filter model.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Scenario of Interference Generating of GNSS-IR, where θ is the elevation angle of the considered GNSS satellite, blue line represents the antenna gain pattern, and h is the antenna effective height. DS is the direct signal and RS the reflected one.

**Figure 2.**Lamasquère Experiment location: (

**a**) map of France with a black box locating the region surrounding Lamasquère; (

**b**) zoom on the Lamaquère plots where the measurements were taken (yellow dot); (

**c**) zoom on the installed instrumentation with 2 ML3 Theta probes and Leica GR25 receiver and AR10 antenna.

**Figure 3.**Data processing workflow: the in situ measurements of soil moisture are used as reference and are compared to the robust regression model and the Robust Kalman Filter model.

**Figure 5.**Estimation results of methods by using amplitude on L2 (PRN1), we highlighted for validation data the main moisture peaks (P1 day 17; P2 day 29) and holes (H1 day 6; H2 day 20; and H3 day 47). We can note that the difference between H1 and P1 is less than 1*10

^{−2}m

^{3}/m

^{3}and between H2 and P2 it is ~1*10

^{−2}m

^{3}/m

^{3}which is low and will allow us to test the precision of the inversions and to show the capacity of this technique to differentiate between very weak soil moisture fluctuations.

**Figure 7.**Estimation results of methods by using phase on L1 (PRN1), we highlighted for validation data the main moisture peaks (P1 day 17; P2 day 29) and holes (H1 day 6; H2 day 20; and H3 day 47). We can note that the difference between H1 and P1 is less than 1*10

^{−2}m

^{3}/m

^{3}and between H2 and P2 it is ~1*10

^{−2}m

^{3}/m

^{3}which is low and will allow us to test the precision of the inversions and to show the capacity of this technique to differentiate between very weak soil moisture fluctuations.

**Figure 8.**Estimation results of dual-band data fusion methods by using amplitude (PRN1), we highlighted for validation data the main moisture peaks (P1 day 17; P2 day 29) and holes (H1 day 6; H2 day 20; and H3 day 47). We can note that the difference between H1 and P1 is less than 1*10

^{−2}m

^{3}/m

^{3}and between H2 and P2 it is ~1*10

^{−2}m

^{3}/m

^{3}which is low and will allow us to test the precision of the inversions and to show the capacity of this technique to differentiate between very weak soil moisture fluctuations.

**Figure 9.**Estimation results of dual-band data fusion methods by using phase, we highlighted for validation data the main moisture peaks (P1 day 17; P2 day 29) and holes (H1 day 6; H2 day 20; and H3 day 47). We can note that the difference between H1 and P1 is less than 1*10

^{−2}m

^{3}/m

^{3}and between H2 and P2 it is ~1*10

^{−2}m

^{3}/m

^{3}which is low and will allow us to test the precision of the inversions and to show the capacity of this technique to differentiate between very weak soil moisture fluctuations.

**Figure 10.**Estimation results of multivariate variable dual-band data fusion methods (PRN1), we highlighted for validation data the main moisture peaks (P1 day 17; P2 day 29) and holes (H1 day 6; H2 day 20; and H3 day 47). We can note that the difference between H1 and P1 is less than 1*10

^{−2}m

^{3}/m

^{3}and between H2 and P2 it is ~1*10

^{−2}m

^{3}/m

^{3}which is low and will allow us to test the precision of the inversions and to show the capacity of this technique to differentiate between very weak soil moisture fluctuations.

Method | L1 | L2 | Dual-Band |
---|---|---|---|

Conventional model | 4 | 14 | 19 |

Robust regression model | 7 | 16 | 19 |

Robust Kalman model | 13 | 18 | 24 |

Method | L1 | L2 | Dual-Band |
---|---|---|---|

Conventional model | 9 | 4 | 12 |

Robust regression model | 12 | 6 | 14 |

Robust Kalman model | 14 | 12 | 18 |

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**MDPI and ACS Style**

Jing, L.; Yang, L.; Yang, W.; Xu, T.; Gao, F.; Lu, Y.; Sun, B.; Yang, D.; Hong, X.; Wang, N.;
et al. Robust Kalman Filter Soil Moisture Inversion Model Using GPS SNR Data—A Dual-Band Data Fusion Approach. *Remote Sens.* **2021**, *13*, 4013.
https://doi.org/10.3390/rs13194013

**AMA Style**

Jing L, Yang L, Yang W, Xu T, Gao F, Lu Y, Sun B, Yang D, Hong X, Wang N,
et al. Robust Kalman Filter Soil Moisture Inversion Model Using GPS SNR Data—A Dual-Band Data Fusion Approach. *Remote Sensing*. 2021; 13(19):4013.
https://doi.org/10.3390/rs13194013

**Chicago/Turabian Style**

Jing, Lili, Lei Yang, Wentao Yang, Tianhe Xu, Fan Gao, Yilin Lu, Bo Sun, Dongkai Yang, Xuebao Hong, Nazi Wang,
and et al. 2021. "Robust Kalman Filter Soil Moisture Inversion Model Using GPS SNR Data—A Dual-Band Data Fusion Approach" *Remote Sensing* 13, no. 19: 4013.
https://doi.org/10.3390/rs13194013