# L-Band SAR Co-Polarized Phase Difference Modeling for Corn Fields

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Incoherent Multi-Parameter Fitting Model

#### 2.2. Sensitivity Analysis of the Model Parameters

- At ${\theta}_{i}={20}^{\xb0}$, the ${\varphi}_{p}$ variations evaluated at the ground measurements are $\Delta {\varphi}_{p}={0}^{\xb0}$ on the $2{a}_{0},h$-space and $\Delta {\varphi}_{p}={2}^{\xb0}$ on ${\epsilon}_{st}^{\prime},{\epsilon}_{st}^{\u2033}$-space;
- Similarly, at ${\theta}_{i}={60}^{\xb0}$ resulted in $\Delta {\varphi}_{p}={26}^{\xb0}$ and $\Delta {\varphi}_{p}={28}^{\xb0}$;
- At both ${\theta}_{i}={20}^{\xb0}$ and ${\theta}_{i}={60}^{\xb0}$, $\Delta {\varphi}_{st}$ is bounded between −6° and −3°.

#### 2.3. Microwave Dielectric Constant of Stalk from Gravimetric Measurements

#### 2.4. Study Area and Ground Data Collection

#### 2.5. SAR Data and Its Quality and Processing Chain

#### 2.6. Polarimetric Observable $\varphi $

## 3. Results

#### 3.1. Co-Polarized Phase Difference ${\varphi}_{0}$ Estimation

#### 3.2. Ulaby’s Model Fitting to SAR Data

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Sensitivity of soil term on the real part of the dielectric constant. The imaginary part is assumed to be 0.10 (black lines) and 0.25 (blue lines) of the real part.

**Figure 2.**Sensitivity analysis of model parameters. (

**a**) Sensitivity of the propagation term on stalk height and diameter $2{a}_{0}$. The ‘+’-mark indicates the average values for the dataset. (

**b**) Sensitivity of propagation term on real ${\epsilon}_{st}^{\prime}$ and imaginary ${\epsilon}_{st}^{\u2033}$ parts of ${\epsilon}_{st}$.

**Figure 4.**MLE fitting for speckled co-polarized phase difference histograms. (

**a**) A 2.27-m-height corn field imaged by UAVSAR at incidence angle 49.98°. (

**b**) A 2.00-m-eight corn field imaged by ALOS-2/PALSAR-2 at incidence angle 26.67°.

**Figure 5.**Model fitting by nonlinear least-squares and estimated parameters. (

**a**) Coherence ${\rho}_{0}$. (

**b**) Co-polarized phase difference ${\varphi}_{0}$ and model fitting. The fitted parameters are indicated. (

**c**) Each contribution to the total phase difference is shown separately.

**Figure 6.**$\varphi $-contours resulting from the evaluation of (2) when coupling Mätzler’s model with Ulaby’s. Frequency is fixed at 1.25 $\mathrm{GHz}$ and incidence angle is 40°. Model parameters now include ${m}_{g}$ instead of ${\epsilon}_{st}$. Stalk height h is indicated. (

**a**) $h=1.80$ $\mathrm{m}$. (

**b**) $h=2.00$ $\mathrm{m}$. (

**c**) $h=2.40$ $\mathrm{m}$. (

**d**) $h=2.80$ $\mathrm{m}$. Contours are in degrees.

**Table 1.**Corn stalk features from the ground data collection for two field campaigns in Canada and Argentina.

Feature | Canada | Argentina |
---|---|---|

(SMAPVex12) | (CONAE) | |

# Fields [-] | 8 | 20 |

# Data points [-] | 32 | 30 |

Stalk height h range [m] | 1.93–2.53 | 1.80–3.00 |

Stalk diameter d range [cm] | 1.85–2.35 | - |

Stalk moisture ${m}_{g}$ range [g/g] | 0.811–0.834 | - |

Stalk density N [1/${\mathrm{m}}^{2}$] | 7.0–8.2 | - |

**Table 2.**Stalk features as compared to the fitting and to the ground data from dates prior to peak biomass. Field campaign in Canada with UAVSAR. The di2electric constant on ground data is estimated from stalk moisture by means of Mätzler’s model shown in Figure 3.

Date | ||||
---|---|---|---|---|

5 July 2012 | 8 July 2012 | 14 July 2012 | 17 July 2012 | |

Fitted pars. | ||||

Height h [m] | 1.42 | 1.83 | 2.56 | 2.60 |

Diameter d [cm] | 1.80 | 1.80 | 1.80 | 1.80 |

Dielectric constant ${\epsilon}_{st}$ [-] | 30.6 + 6.0i | 31.4 + 6.0i | 32.0 + 6.0i | 24.9 + 6.0i |

Density N [1/m^{2}] | 7.15 | 7.39 | 8.16 | 8.20 |

Root mean sq. error [°] | 16.3 | 20.8 | 21.8 | 22.3 |

Ground data | ||||

Height h range [m] | 1.19–1.77 | 1.93–2.53 | ||

Diameter d range [cm] | 2.00–2.29 | 1.85–2.35 | ||

Moisture ${m}_{g}$ range [g/g] | 0.834–0.847 | 0.811–0.834 | ||

Dielectric constant (real part) ${\epsilon}_{st}^{\prime}$ [-] | 32–34 | 31–32 |

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**MDPI and ACS Style**

Barber, M.E.; Rava, D.S.; López-Martínez, C.
L-Band SAR Co-Polarized Phase Difference Modeling for Corn Fields. *Remote Sens.* **2021**, *13*, 4593.
https://doi.org/10.3390/rs13224593

**AMA Style**

Barber ME, Rava DS, López-Martínez C.
L-Band SAR Co-Polarized Phase Difference Modeling for Corn Fields. *Remote Sensing*. 2021; 13(22):4593.
https://doi.org/10.3390/rs13224593

**Chicago/Turabian Style**

Barber, Matías Ernesto, David Sebastián Rava, and Carlos López-Martínez.
2021. "L-Band SAR Co-Polarized Phase Difference Modeling for Corn Fields" *Remote Sensing* 13, no. 22: 4593.
https://doi.org/10.3390/rs13224593