# Uncertainties and Perspectives on Forest Height Estimates by Sentinel-1 Interferometry

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Sentinel 1 Data

#### 2.2. Interferometric Phase Modelling

#### 2.3. Modelling dh Uncertainty

_{i}):

#### 2.3.1. Theoretical Uncertainty of ω

#### 2.3.2. Theoretical Uncertainty of $d\mathsf{\Delta}\phi $

#### 2.4. Minimizing ${\sigma}_{dh}$ through Simulated Scenarios

_{v}values (i.e., expected average forest height). With reference to the scenarios, the optimal B value can be retrieved once an expected forest h

_{v}is set. To make the estimate more immediate, a power model (Equation (13)) was calibrated directly relating h

_{v}with the optimal B value:

_{v}were found, that part of ${\sigma}_{dh}$, depending on settable operational parameters, can be finally minimized.

## 3. Results and Discussion

#### 3.1. Theoretical Uncertainty of $\omega $

#### 3.2. Theoretical Uncertainty of $d\Delta \phi $

_{v}participates to reduce ${\gamma}_{vol}$. Figure 4b shows a perfect negative linear correlation between B and ${\gamma}_{geom}$ having a steeper decreasing rate for lower look angles. Because ${\gamma}_{baseline}={\gamma}_{geom}\xb7{\gamma}_{vol}$, according to Equation (9), one can admit that the higher the baseline, the lower ${\gamma}_{baseline}$ and the higher ${\sigma}_{\mathsf{\Delta}\phi}$.

#### 3.3. Minimizing ${\sigma}_{dh}$ through Simulated Scenarios

_{v}. To explore its dependency, B and h

_{v}were changed progressively from 5 m to 1000 m and from 5 m to 50 m, respectively, testing their effects on ${\sigma}_{dh}$ (Figure 5a).

_{v}value > 15 m (i.e., the majority of forests), and ${\sigma}_{dh}$ presents a minimum with respect to B.

_{v}values, a model directly relating the “optimal” B value with the expected h

_{v}was defined (Equation (13) and Figure 5b).

^{2}coefficients). It is worth noting that the optimal B value occurs within the critical baseline (for S1, about 5 km) supporting the hypothesis that, over vegetation, the accuracy of interferometric-derived heights does not increase by using large baselines. The operational utility of this model can be easily exemplified using a case study. Suppose we investigate tree heights in a forest having an expected value of 25 m. The model of Figure 5b makes possible to obtain an optimal baseline value of 150 m. Similarly, it can be said that the optimal B value in forests with tree heights ranging between 15 m and 30 m (the majority of forest in temperate zones) should range between 250 m and 100 m, respectively.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Workflow adopted in this work. B is the baseline; NL is the interferogram multilooking factor; h

_{v}is the expected average forest height; dh is the estimated forest height according to proposed model; $\mathsf{\Delta}{\phi}^{FP}$ and $\mathsf{\Delta}{\phi}^{RP}$ are the interferometric phases of a forest point and a reference (ground) point, respectively; $\omega $ is the gain factor that allows the conversion of an interferometric phase difference ($d\mathsf{\Delta}\phi $) into height difference.

**Figure 2.**Maximum detectable dh (HOA), avoiding phase unwrapping, versus B and θ (simulation performed with reference to S1 nominal features).

**Figure 3.**(

**a**) ${\sigma}_{\omega}$ versus B and θ. (

**b**) relative weights affecting $\omega $ variance involved in VPL. Note that w

_{B}is exaggerated by a factor 200.

**Figure 5.**(

**a**) Several scenarios of dh uncertainty according to h

_{v}and B. (

**b**) Power model relating h

_{v}and B at dh uncertainty is minimized.

**Figure 6.**SGRP (squared pixel size) vs. coherence. A ${\sigma}_{dh}$ equal to 3 m was used during simulations obtained by varying γ and B.

**Figure 7.**Several scenarios of dh uncertainty according to expected ${\gamma}_{FP}$ and dh at different B values. Isolines refer to the same ${\sigma}_{dh}$ values.

Feature | Values | Units |
---|---|---|

Frequency (λ) | 5.54 | cm |

Nominal Satellite Altitude (H) | 693 | km |

Look Angle (θ) | 30–45 | ° |

$\mathrm{Attitude}\mathrm{accuracy}({\sigma}_{\theta}$) | 0.01 | ° |

Maximum Noise Equivalent Sigma Zero (NESZ) | −22 | dB |

$\mathrm{Spatial}\mathrm{resolution}\mathrm{range}({\delta}_{rg}$) | 5 | m |

$\mathrm{Spatial}\mathrm{resolution}\mathrm{azimuth}({\delta}_{az}$) | 20 | m |

Satellite position accuracy POD | 5 | cm |

Bandwidth (Bw) | 42–56 | MHz |

Antenna real length (L) | 12 | m |

**Table 2.**“Weights” defining the relative importance of factors to determine ${\sigma}_{\omega}^{2}$.

Parameter | ${\mathit{w}}_{\mathit{i}}$Formula |
---|---|

Baseline (B) | ${w}_{B}=\frac{{\left(\frac{\partial \omega}{\partial B}\right)}^{2}\xb7{\sigma}_{B}^{2}}{{\sigma}_{\omega}{}^{2}}$ |

Slant range (R) | ${w}_{R}=\frac{{\left(\frac{\partial \omega}{\partial R}\right)}^{2}\xb7{\sigma}_{R}^{2}}{{\sigma}_{\omega}{}^{2}}$ |

Look angle (θ) | ${w}_{\theta}=\frac{{\left(\frac{\partial \omega}{\partial \theta}\right)}^{2}\xb7{\sigma}_{\theta}^{2}}{{\sigma}_{\omega}{}^{2}}$ |

Mixed term (R, θ) | ${w}_{corr\left(R,\theta \right)}=\frac{\left(\frac{\partial \omega}{\partial R}\frac{\partial \omega}{\partial \theta}\right)\xb7{\rho}_{\left(R,\theta \right)}{\sigma}_{R}{\sigma}_{\theta}}{{\sigma}_{\omega}{}^{2}}$ |

**Table 3.**Best cases from simulations in a typical Italian forest context (tree height in the range of 10–30 m).

Baseline (m) | Expected dh (m) | ${\mathit{\sigma}}_{\mathit{d}\mathit{h}}$(m) |
---|---|---|

50 | 10–30 | 2 |

100 | 10–30 | 2 |

150 | 10–30 | 1 |

200 | 10–30 | 0.5 |

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**MDPI and ACS Style**

De Petris, S.; Sarvia, F.; Borgogno-Mondino, E.
Uncertainties and Perspectives on Forest Height Estimates by Sentinel-1 Interferometry. *Earth* **2022**, *3*, 479-492.
https://doi.org/10.3390/earth3010029

**AMA Style**

De Petris S, Sarvia F, Borgogno-Mondino E.
Uncertainties and Perspectives on Forest Height Estimates by Sentinel-1 Interferometry. *Earth*. 2022; 3(1):479-492.
https://doi.org/10.3390/earth3010029

**Chicago/Turabian Style**

De Petris, Samuele, Filippo Sarvia, and Enrico Borgogno-Mondino.
2022. "Uncertainties and Perspectives on Forest Height Estimates by Sentinel-1 Interferometry" *Earth* 3, no. 1: 479-492.
https://doi.org/10.3390/earth3010029