**3.4.3 Parametric tomographic analysis**

14 Will-be-set-by-IN-TECH

azimuth). The analyzed area corresponds to a stripe of the data along the slant range direction, shown in the top panel of Fig. (7). Almost the whole stripe is forested except for the dark areas in near range, corresponding to bare terrain. It may be observed that the three spectra have very similar characteristics. Each spectrum is characterized by a narrow peak, above which a weak sidelobe is visible. As for the co-polar channels, this result is consistent with the hypothesis that a single scattering mechanism is dominant. It is then reasonable to relate such scattering mechanism to the double bounce contribution from trunk-ground and canopy-ground interactions, and hence the peak of the spectrum can be assumed to be located at ground level. The sidelobe above the main peak is more evident in the HV channel, but the contributions from ground level seem to be dominant as well, the main peak being located almost at the same position as in the co-polar channels. Accordingly, the presence of a

Capon Spectrum - HH

0 500 1000 1500 2000

Mean Reflectivity (HH)

0 500 1000 1500 2000

0 500 1000 1500 2000

Capon Spectrum - VV

slant range [m] 0 500 1000 1500 2000

Fig. 7. Top panel: mean reflectivity of the data (HH channel) within a stripe as wide as 50 m in the azimuth direction. The underlying panels show the Capon Spectra for the three polarimetric channels. At every range bin the signal has been scaled in such a way as to have

Capon Spectrum - HV

relevant contribution from the ground has to be included in the HV channel too.

elevation [m]

azimuth [m]

10

30

50

elevation [m]

elevation [m]

unitary energy.

Results shown here have been obtained by processing all polarizations at once, as suggested in section 3.3.1.

Figure (8) shows the map of the estimates relative ground elevation. The black areas correspond to absence of coherent signals, as it is the case of lakes. The estimates relative to canopy elevation are visible in Fig. (9). In this case, the black areas have been identified by the algorithm as being non-forested. It is worth noting the presence of a road, clearly visible in the optical image, see Fig. (5), crossing the scene along the direction from slant range, azimuth coordinates (1850, 0) to (1000, 5500). Along that road, a periodic series of small targets at an elevation of about 25 m has been found by the algorithm. Since a power line passes above the road, it seems reasonable to relate such targets to the echoes from the equipment on the top of the poles of the power line.

Fig. 8. Top row: ground elevation estimated by LIDAR. Bottom row: ground elevation estimated by T-SAR. Black areas correspond to an unstructured scattering mechanism.

As a validation tool, we used LIDAR measurements courtesy of the Swedish Defence Research Agency (FOI) and Hildur and Sven Wingquist's Foundation. Concerning ground elevation the dispersion of the difference *zSAR* − *zLIDAR* has been assessed in less than 1 m. Concerning the estimated canopy elevation, the discrepancy with respect to LIDAR is clearly imputable to the fact that the estimates are relative to the average the *phase center* elevation inside the estimation window, whereas LIDAR is sensitive to the top height of the canopy. In particular, canopy elevation provided by T-SAR appears to be under-estimated with respect to LIDAR measurements, as a result of the under foliage penetration capabilities of P-band microwaves. Figure (10) reports the ratios between the estimated backscattered powers from the ground and the canopy (G/C ratio), for each polarimetric channel. As expected, in the co-polar channels the scattered power from the ground is significantly larger than canopy backscatter, the G/C ratio being assessed in about 10 dB. In the HV channel the backscattered powers

Multi-Baseline SARs 17

Forest Structure Retrieval from Multi-Baseline SARs 43

HH – Ground to Canopy Ratio [dB]

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500

HV – Ground to Canopy Ratio [dB]

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500

VV – Ground to Canopy Ratio [dB]

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500

presence of a dihedral contribution. Now, whereas for a perfect conducting dihedral Δ*ϕ* is exactly 180◦, for a lossy dielectric dihedral a lower value of Δ*ϕ* is expected, due to the electromagnetic properties of the trunk-ground ensemble (in (37) for example, a co-polar phase of 94◦ has been observed for trunk-ground scattering). What makes the interpretation of the co-polar signature of forested areas not straightforward is that low values of the co-polar coherence have been observed, whereas dihedral contributions to the HH and VV channels are usually assumed to by highly correlated (36). A possible explanation would be to assume that forested areas are characterized by an almost ideal dihedral scattering plus a significant contribution from canopy backscatter, which would explain both the low co-polar coherence and the reduction of Δ*ϕ* from the ideal value of 180◦ to 80◦. This interpretation, however, is not satisfying, since it is not consistent with the presence of highly amplitude stable targets nor with the Capon spectra in Fig. (7). At this point, a better explanation seems to be that to consider forested areas as being dominated by a *single* scattering mechanism, responsible for (most of) the coherence loss between the co-polar channels, highly coherent with respect to geometrical and temporal variations, and approximately located at ground level. From a physical point of view, it makes sense to relate such scattering mechanism to trunk-ground interactions, eventually perturbed by the presence of understory, trunk and ground roughness, small oscillations in the local topography, or eventually by canopy-ground interactions. Casting T-SAR as a parametric estimation problem has allowed to support such a conclusion by providing quantitative arguments such as the G/C ratios that indicate that not only in the co-polar channel, but also in the HV channel the contributions from the ground level dominate those from the canopy. Such result indicates that the ideal dihedral scattering model does not provide a sufficient description of the HV ground contributions under the

azimuth [m]

Fig. 10. Ground to Canopy Ratio for the three polarimetric channels.

forest, suggesting effects due to understory and volume-ground interactions.

slant range [m]

slant range [m]

slant range [m]

0

0

0

1000

2000

1000

2000

1000

2000

Fig. 9. Top row: canopy elevation estimated by LIDAR. Bottom row: canopy elevation estimated by T-SAR. Black areas correspond to absence of canopy.

from the ground and the canopy are closer to each other, even though ground contributions still appear to be dominant, resulting in a ground to canopy ratio of about 3 dB.

#### 3.4.3.1 Single-polarization analysis

This last paragraph is dedicated to reporting the results relative to the elevation estimates provided by processing the HH, VV, and HV channel separately, compared to estimates yielded by the fully polarimetric (FP) tomography commented above. The joint distributions of the elevation estimates yielded by the FP tomography and by the single channel tomographies are reported in Fig. (11). It can be appreciated that, as expected, ground elevation is better estimated by processing the co-polar channel, whereas canopy elevation is better estimated by processing the HV channel. In all cases, however, the estimates are close to those provided by the FP tomography, proving that the tomographic characterization of forested areas may be carried out on the basis of a single polarimetric channel, provided that a sufficient number of acquisitions is available. It is important to note that the canopy phase center elevation yielded by the VV tomography has turned out to be slightly (between 1 and 2 meters) over estimated, with respect to both the FP tomography and to the HH and HV tomographies. This phenomenon indicates that the scatterers within the canopy volume exhibit a vertical orientation, as expected at longer wavelengths (34), (35). This result shows that effects of vegetation orientation at P-band are appreciable, and measurable by Tomography, but not so severe to hinder the applicability of the FP Tomography depicted above.

#### **3.4.4 Discussion**

Since the co-polar phase for both ground and canopy backscatter is expected to be null, (36), the value of Δ*ϕ* ≈ 80◦ found in forested areas can only be interpreted as an index of the 16 Will-be-set-by-IN-TECH

LIDAR: Canopy Elevation

azimuth [m]

Fig. 9. Top row: canopy elevation estimated by LIDAR. Bottom row: canopy elevation

still appear to be dominant, resulting in a ground to canopy ratio of about 3 dB.

estimated by T-SAR. Black areas correspond to absence of canopy.

500 1000 1500 2000 2500 3000 3500 4000

from the ground and the canopy are closer to each other, even though ground contributions

This last paragraph is dedicated to reporting the results relative to the elevation estimates provided by processing the HH, VV, and HV channel separately, compared to estimates yielded by the fully polarimetric (FP) tomography commented above. The joint distributions of the elevation estimates yielded by the FP tomography and by the single channel tomographies are reported in Fig. (11). It can be appreciated that, as expected, ground elevation is better estimated by processing the co-polar channel, whereas canopy elevation is better estimated by processing the HV channel. In all cases, however, the estimates are close to those provided by the FP tomography, proving that the tomographic characterization of forested areas may be carried out on the basis of a single polarimetric channel, provided that a sufficient number of acquisitions is available. It is important to note that the canopy phase center elevation yielded by the VV tomography has turned out to be slightly (between 1 and 2 meters) over estimated, with respect to both the FP tomography and to the HH and HV tomographies. This phenomenon indicates that the scatterers within the canopy volume exhibit a vertical orientation, as expected at longer wavelengths (34), (35). This result shows that effects of vegetation orientation at P-band are appreciable, and measurable by Tomography, but not so severe to hinder the applicability of the FP Tomography depicted

Since the co-polar phase for both ground and canopy backscatter is expected to be null, (36), the value of Δ*ϕ* ≈ 80◦ found in forested areas can only be interpreted as an index of the

SAR: Canopy Elevation

500 1000 1500 2000 2500 3000 3500 4000

slant range [m]

slant range [m]

3.4.3.1 Single-polarization analysis

above.

**3.4.4 Discussion**

HH – Ground to Canopy Ratio [dB]

Fig. 10. Ground to Canopy Ratio for the three polarimetric channels.

presence of a dihedral contribution. Now, whereas for a perfect conducting dihedral Δ*ϕ* is exactly 180◦, for a lossy dielectric dihedral a lower value of Δ*ϕ* is expected, due to the electromagnetic properties of the trunk-ground ensemble (in (37) for example, a co-polar phase of 94◦ has been observed for trunk-ground scattering). What makes the interpretation of the co-polar signature of forested areas not straightforward is that low values of the co-polar coherence have been observed, whereas dihedral contributions to the HH and VV channels are usually assumed to by highly correlated (36). A possible explanation would be to assume that forested areas are characterized by an almost ideal dihedral scattering plus a significant contribution from canopy backscatter, which would explain both the low co-polar coherence and the reduction of Δ*ϕ* from the ideal value of 180◦ to 80◦. This interpretation, however, is not satisfying, since it is not consistent with the presence of highly amplitude stable targets nor with the Capon spectra in Fig. (7). At this point, a better explanation seems to be that to consider forested areas as being dominated by a *single* scattering mechanism, responsible for (most of) the coherence loss between the co-polar channels, highly coherent with respect to geometrical and temporal variations, and approximately located at ground level. From a physical point of view, it makes sense to relate such scattering mechanism to trunk-ground interactions, eventually perturbed by the presence of understory, trunk and ground roughness, small oscillations in the local topography, or eventually by canopy-ground interactions. Casting T-SAR as a parametric estimation problem has allowed to support such a conclusion by providing quantitative arguments such as the G/C ratios that indicate that not only in the co-polar channel, but also in the HV channel the contributions from the ground level dominate those from the canopy. Such result indicates that the ideal dihedral scattering model does not provide a sufficient description of the HV ground contributions under the forest, suggesting effects due to understory and volume-ground interactions.

Multi-Baseline SARs 19

Forest Structure Retrieval from Multi-Baseline SARs 45

assumption of two SMs matches the intuitive argument that a forested area is characterized by the presence of two objects, i.e.: the ground and the vegetation layer. This idea has been formalized in literature through different physical models, considering the features of ground and volume scattering in polarimetric data (38), single-polarization tomographic data, as shown in (26) and in the previous section, or in polarimetric and interferometric (PolInSAR) data (13), (39), (27). Beside physical soundness, however, the popularity of two-layered models is also due to the fact that they provide a sufficiently simple mathematical framework to allow model inversion. This is particularly important in PolInSAR analysis, where the assumption of two layers results in the coherence loci, namely the distribution of the interferometric coherence as a function of polarization, to be given by a straight line in the complex plane (39). This simple geometrical interpretation provides the key to decompose the interferometric coherence in ground-only and volume-only contributions, after which ground and volume parameters, like terrain topographic and canopy heights are retrieved. The analysis of the shape of the coherence loci also provides a direct idea about the soundness of approximating the scene as being constituted by two SMs, which allows to assess the impact of model mismatches (35). The Sum of Kronecker Product (SKP) structure has been proposed in (40) as a general framework to discuss problem inversion in both single and multi-baseline configurations, and independently on the particular physical model adopted to represent each SM. Concerning two-layered models, the SKP formalism leads to the conclusion that the correct identification of the structural and polarimetric properties of ground and volume scattering is subject to an ambiguity, in that different solutions exist that fit the data covariance matrix up to the same error. Such an ambiguity is shown to be completely described by two degrees of freedom, which can be resolved by employing physical models. In other words, the two dimensional ambiguity following after the SKP structure represents exactly the model space, meaning that a certain physical model corresponds to a certain solution of problem ambiguity, and vice-versa. Accordingly, the SKP formalism provides a way to discuss every possible physical model, by exploring the space of ambiguous solutions. In this chapter, this methodology is applied to data from the ESA campaigns BIOSAR 2007, BIOSAR 2008 and TROPISAR. Different models are being investigated by exploring different solutions in the ambiguous space, whose features are discussed basing on polarimetric and tomographic

We consider a scenario where Radar scattering is contributed by multiple Scattering Mechanisms (SMs), as in forested areas, and assume a data-set of *N* · *Np* SAR SLC images, *Np* being the number of independent polarizations (typically *Np* = 3 ) and *N* the number of passages over the scene. Let *yn* (**w***i*) denote a complex-valued pixel of the image acquired from passage *n* in the polarization identified by the projection vector **w***i*. A simple way to model the data second order statistics is to assume that: i) different SMs are uncorrelated with one another; ii) the correlation between any two passages, say *n* and *m*, of the *k-th* SM alone, *rk* (*n*, *m*), is invariant to polarization (up to a scale factor); iii) the correlation between any two

passage (up to a scale factor), see (40) for details. Under the three hypotheses above it follows

*K* ∑ *k*=1 *ck* **w***i*, **w***<sup>j</sup>* 

 **w***i*, **w***<sup>j</sup>* 

*,* is invariant to the choice of the

*rk* (*n*, *m*) ∀**w***i*, **w***j*, *n*, *m* (25)

features.

that:

**4.2 The SKP structure**

polarizations, say **w***<sup>i</sup>* and **w***j*, of each SM *alone, ck*

*yn* (**w***i*) *y*<sup>∗</sup> *m* **w***j* <sup>=</sup>

*E* 

Fig. 11. Joint distribution of the phase center elevation estimates yielded by processing the single channels separately (vertical axis) and by the best tomography (horizontal axis). The black line denotes the ideal linear trend. The color scale is proportional to the natural logarithm of the number of counts within each bin.
