**4.5 Physical interpretation**

22 Will-be-set-by-IN-TECH

Branch Boundary Rank-deficient matrix

(29)

*a* Inner **C***<sup>v</sup> a* Outer **R***<sup>g</sup> b* Inner **C***<sup>g</sup> b* Outer **R***<sup>v</sup>*

Table 1. Rank deficiencies at the boundaries of the region of positive definitiveness.

Fig. 12. Regions of physical validity for the interferometric coherences associated with ground and volume scattering in the interferometric pair between the first two tracks of the

coherences associated with ground and volume scattering in the considered interferometric pair. The red and green segments denote the set of all physically valid solutions obtained by

one that corresponds to the true<sup>4</sup> ground and volume coherences, whereas the region of physical validity can be simply associated with two segments along the line passing through the true ground and volume coherences, see figure 12. By definition, the points outer or inner boundaries of the two segments correspond to the case where one of the four matrices {**C***k*, **R***k*}*k*=*g*,*<sup>v</sup>* in equations (28) is singular. In particular, the outer boundaries correspond to rank-deficient structure matrices, whereas the inner boundaries correspond to rank-deficient polarimetric signature, as reported in Table 1. In the single baseline case (*N* = 2) the points at the outer boundary of both segments belong to the unit circle, indicating that physically valid ground and volume interferometric coherences are allowed be unitary in magnitude, see figure 12, left panel. This conclusion is exactly the same as the one drawn in (39), after which it follows the consistency of the SKP formalism with respect to PolInSAR. As new acquisitions are gathered, instead, the positive definitiveness constraint results in the regions of physical validity to shrink from the outer boundaries towards the true ground and volume coherences, whereas the position of the inner boundary points stay unvaried for the considered interferogram, figure 12, middle and right panels. Accordingly, the availability of multiple-baseline results not only in enhanced vertical resolution capabilities, but also in the

data-set. *N* is the total number of available tracks exploited to enforce the positive definitiveness constraint. The black and blue points denote the true interferometric

varying *a* and *b*, respectively.

progressive elimination of incorrect solutions.

<sup>4</sup> Assuming (27).

A straightforward physical interpretation of the polarimetric and structural properties of models associated with different solutions can be provided by analyzing the inner and outer boundaries of the RPV:


#### **4.6 Case studies**

We present here experimental results relative to three case studies based on data from the ESA campaigns BIOSAR 2007 (45), BIOSAR 2008 (46) and TROPISAR. The main features of the analyzed data are reported in tables 2, 3, 4.

Multi-Baseline SARs 25

Forest Structure Retrieval from Multi-Baseline SARs 51

Fig. 13. Tomographic profiles along an azimuth cut corresponding to three different

sensitivity of the data to both wavelength and forest structure.

scattering is not supposed to occur, we conclude that depolarized contributions from the ground level are present in all data-sets. The extent of such contributions is by far more relevant in the case of the BioSAR 2007 and BioSAR 2008 P-Band data-sets, witnessing the

• The position along the vertical axis at which the backscattered power drops down is independent of the choice of the solution, meaning that all solutions carry the same

The figure below reports the amount of information carried by the data that is correctly represented by assuming the data covariance matrix is a sum of 1 to 4 KPs. The measurement

physically valid models for the volume structure.

information about forest top height.

**4.6.2 Fitness**


Table 3. The BioSAR 2008 data-set


Table 4. The TropiSAR data-set

All of the result to follow have been obtained by estimating the sample covariance matrix of the multi-baseline and multi-polarimetric data by multi-looking over a 60 × 60 m (ground range, azimuth) estimation window.

#### **4.6.1 Models for the volume structure**

Figures from13 to 16 report tomographic images of the volume structures as obtained by taking three different solutions on branch *b*. The tomographic imaging has been performed by employing the Capon Spectral Estimator.

All the tomographic profiles behave consistently with the physical features discussed in the previous section. In particular, all profiles associated with the outer boundary solution result in very thin volumes at a high elevation, therefore accounting for the upper vegetation layer only. As the solution is moved towards the inner boundary contributions from the lower vegetation layer appear. When the inner boundary is reached the the contributions appear from the ground level up to top forest height. The following points are worth noting:

• Contributions from the ground level are observed in all cases in the inner boundary solution. As this solution corresponds by construction to the polarization where ground 24 Will-be-set-by-IN-TECH

Site Krycklan river catchment, Northern

Sweden

North-East) – Fully Polarimetric

Campaign BioSAR 2008 Acquisition System E-SAR - DLR Acquisition Period Fall 2008

Scene Boreal forest Topography Hilly

Tomographic Tracks 6 per flight direction (South-West and

Band P-Band and L-Band

Vertical resolution at L-Band 6 m (near range) to 40 m (far range) Vertical resolution at P-Band 20 m (near range) to >80 m (far range)

> Campaign TropiSAR Acquisition System Sethi- ONERA Acquisition Period August 2009

Topography Flat Tomographic Tracks 6 - Fully Polarimetric Band P-Band

All of the result to follow have been obtained by estimating the sample covariance matrix of the multi-baseline and multi-polarimetric data by multi-looking over a 60 × 60 m (ground

Figures from13 to 16 report tomographic images of the volume structures as obtained by taking three different solutions on branch *b*. The tomographic imaging has been performed

All the tomographic profiles behave consistently with the physical features discussed in the previous section. In particular, all profiles associated with the outer boundary solution result in very thin volumes at a high elevation, therefore accounting for the upper vegetation layer only. As the solution is moved towards the inner boundary contributions from the lower vegetation layer appear. When the inner boundary is reached the the contributions appear

• Contributions from the ground level are observed in all cases in the inner boundary solution. As this solution corresponds by construction to the polarization where ground

from the ground level up to top forest height. The following points are worth noting:

Slant Range resolution 1 m Azimuth resolution 1 m Vertical resolution �15 m

Site Paracou, French Guyana Scene Tropical forest

Slant Range resolution 1.5 m Azimuth resolution at L-Band 1.2 m Azimuth resolution at P-Band 1.6 m

Table 3. The BioSAR 2008 data-set

Table 4. The TropiSAR data-set

range, azimuth) estimation window.

**4.6.1 Models for the volume structure**

by employing the Capon Spectral Estimator.

Fig. 13. Tomographic profiles along an azimuth cut corresponding to three different physically valid models for the volume structure.

scattering is not supposed to occur, we conclude that depolarized contributions from the ground level are present in all data-sets. The extent of such contributions is by far more relevant in the case of the BioSAR 2007 and BioSAR 2008 P-Band data-sets, witnessing the sensitivity of the data to both wavelength and forest structure.

• The position along the vertical axis at which the backscattered power drops down is independent of the choice of the solution, meaning that all solutions carry the same information about forest top height.

#### **4.6.2 Fitness**

The figure below reports the amount of information carried by the data that is correctly represented by assuming the data covariance matrix is a sum of 1 to 4 KPs. The measurement

Multi-Baseline SARs 27

Forest Structure Retrieval from Multi-Baseline SARs 53

Fig. 15. Tomographic profiles along an azimuth cut corresponding to three different physically valid models for the volume structure. Note that topography has been removed

• The assumption of 2 KPs has turned out to account for about 90% of the information carried by the data in all investigated cases, meaning that two-layered models (ground plus volume) are well suited for forestry investigations. Accounting for other components, such as subsurface penetration or differential extinction phenomena, does not appear to be necessary as for a first-order characterization of the forest structure, their role being limited

On the basis of the analyzed data-sets, the following conclusions are drawn:

such that the ground level is always at 0 m.

to about 10% of the total information content.

**4.7 Discussion**

Fig. 14. Tomographic profiles along an azimuth cut corresponding to three different physically valid models for the volume structure. Note that topography has been removed such that the ground level is always at 0 m.

is carried out as *IK* = 1 − *εK*, *ε<sup>K</sup>* being defined as:

$$\varepsilon\_{K} = \frac{\left\| \widehat{\mathbf{W}} - \widehat{\mathbf{W}}\_{K} \right\|\_{F}}{\left\| \widehat{\mathbf{W}} \right\|\_{F}}$$

where **W** is the sample covariance matrix and **W***<sup>K</sup>* its best estimate in the Frobenius norm by using *K* KPs, see (40) for details.

26 Will-be-set-by-IN-TECH

Fig. 14. Tomographic profiles along an azimuth cut corresponding to three different physically valid models for the volume structure. Note that topography has been removed

*ε<sup>K</sup>* =

 **<sup>W</sup>** <sup>−</sup> **<sup>W</sup>***<sup>K</sup>*

 **W** *F*

where **W** is the sample covariance matrix and **W***<sup>K</sup>* its best estimate in the Frobenius norm by

 *F*

such that the ground level is always at 0 m.

using *K* KPs, see (40) for details.

is carried out as *IK* = 1 − *εK*, *ε<sup>K</sup>* being defined as:

Fig. 15. Tomographic profiles along an azimuth cut corresponding to three different physically valid models for the volume structure. Note that topography has been removed such that the ground level is always at 0 m.

#### **4.7 Discussion**

On the basis of the analyzed data-sets, the following conclusions are drawn:

• The assumption of 2 KPs has turned out to account for about 90% of the information carried by the data in all investigated cases, meaning that two-layered models (ground plus volume) are well suited for forestry investigations. Accounting for other components, such as subsurface penetration or differential extinction phenomena, does not appear to be necessary as for a first-order characterization of the forest structure, their role being limited to about 10% of the total information content.

Multi-Baseline SARs 29

Forest Structure Retrieval from Multi-Baseline SARs 55

Fig. 17. Information content represented by taking *K* KPs for the four analyzed data-sets.

physically valid data-consistent models.

largely independently on baseline aperture.

**5. Conclusions**

decorrelation.

from multi-baseline SARs.

On the other hand, though, retrieving the correct forest height does not provide arguments to assess the validity of the whole model, as the same top height is consistent with different

This chapter has considered the retrieval of information about the forest vertical structure

Considering a purely tomographic formulation of the problem, it has been shown that the backscattered power associated with a certain depth within the vegetation layer can be retrieved with a *virtually* arbitrary resolution, simply by employing a sufficiently large baseline apertures. Besides costs, however, baseline aperture is upper bounded by physical constraints arising from the nature of the targets themselves, such as anisotropy and temporal

Posing T-SAR as an estimation problems greatly helps compensate for the coarse resolution arising from an insufficient baseline aperture, allowing to retrieve useful information about the forest structure even in cases where the Fourier resolution is many times the overall forest height. In particular, T-SAR has been shown to be capable of identifying bald and forested areas and estimating the elevation and backscattered power of the scattering centers representing associated with ground and volume scattering. Furthermore, it has been shown that T-SAR can be employed basing on both single and multi-polarimetric data, which makes T-SAR a valuable tool to investigate variations of the forest structure with polarization. The availability of multi-polarimetric and multi-baseline data has been shown to provide the most information, allowing the decomposition of the data covariance matrix into ground-only and volume-only contributions even in absence of a parametric model and

The capability of the SKP formalism to represent *all* physically valid and data-consistent two-layered models has allowed an exhaustive discussion about the validity of such a class of models for the analysis of forested areas. As a result, it has been observed that two SMs account for more than 90% of the information carried by the data in all investigated cases.

Fig. 16. Tomographic profiles along an azimuth cut corresponding to three different physically valid models for the volume structure. Note that topography has been removed such that the ground level is always at 0 m.


Fig. 17. Information content represented by taking *K* KPs for the four analyzed data-sets.

On the other hand, though, retrieving the correct forest height does not provide arguments to assess the validity of the whole model, as the same top height is consistent with different physically valid data-consistent models.

#### **5. Conclusions**

28 Will-be-set-by-IN-TECH

Fig. 16. Tomographic profiles along an azimuth cut corresponding to three different physically valid models for the volume structure. Note that topography has been removed

• The assumption of a ground-free polarization has resulted in an evenly distributed volume structure in a boreal forest at L-Band and in a tropical forest at P-Band, and in an almost ground-locked volume structure in a boreal and semi-boreal forest at P-Band, witnessing the sensitivity of Radar data to wavelength and forest structure. The extent of depolarized contributions from the ground level suggests ground-volume interaction phenomena may occur at P-Band in sparse forests. If this is the case, volume backscattering can be retrieved

• The retrieval of forest top height is nearly invariant to the choice of the model. In this sense, forest top height appears as the most robust indicator of the forest structure as observed through microwaves measurements, providing a further and independent argument supporting the validity of PolInSAR for the remote sensing of forested scenarios.

such that the ground level is always at 0 m.

by allowing ground scattering to be partly entropic.

This chapter has considered the retrieval of information about the forest vertical structure from multi-baseline SARs.

Considering a purely tomographic formulation of the problem, it has been shown that the backscattered power associated with a certain depth within the vegetation layer can be retrieved with a *virtually* arbitrary resolution, simply by employing a sufficiently large baseline apertures. Besides costs, however, baseline aperture is upper bounded by physical constraints arising from the nature of the targets themselves, such as anisotropy and temporal decorrelation.

Posing T-SAR as an estimation problems greatly helps compensate for the coarse resolution arising from an insufficient baseline aperture, allowing to retrieve useful information about the forest structure even in cases where the Fourier resolution is many times the overall forest height. In particular, T-SAR has been shown to be capable of identifying bald and forested areas and estimating the elevation and backscattered power of the scattering centers representing associated with ground and volume scattering. Furthermore, it has been shown that T-SAR can be employed basing on both single and multi-polarimetric data, which makes T-SAR a valuable tool to investigate variations of the forest structure with polarization.

The availability of multi-polarimetric and multi-baseline data has been shown to provide the most information, allowing the decomposition of the data covariance matrix into ground-only and volume-only contributions even in absence of a parametric model and largely independently on baseline aperture.

The capability of the SKP formalism to represent *all* physically valid and data-consistent two-layered models has allowed an exhaustive discussion about the validity of such a class of models for the analysis of forested areas. As a result, it has been observed that two SMs account for more than 90% of the information carried by the data in all investigated cases.

Multi-Baseline SARs 31

Forest Structure Retrieval from Multi-Baseline SARs 57

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The correct identification of physical models for such two SMs has been shown to be subject to an ambiguity, mostly associated with the possibility that depolarized contributions occur at the ground level as well. Furthermore, the correct retrieval of volume top height has been shown not to constitute a meaningful tool for model validation, being substantially invariant to the choice of the solution for volume scattering. In this framework it is then clear that tomographic imaging represents a most valuable tools for the assesment of physical models of the forest structure, as it allows to *see* what kind of vertical structure is actually associated with the chosen model.
