**4. Ground-volume decomposition from multi-baseline and multi-polarimetric data**

#### **4.1 Introduction**

The idea that Radar scattering from forested areas can be well modeled as being constituted by two Scattering Mechanisms (SMs) has largely been retained in literature. In first place, the 18 Will-be-set-by-IN-TECH

VV Tomography

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

**4. Ground-volume decomposition from multi-baseline and multi-polarimetric data**

The idea that Radar scattering from forested areas can be well modeled as being constituted by two Scattering Mechanisms (SMs) has largely been retained in literature. In first place, the

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**4.1 Introduction**

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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 features.

#### **4.2 The SKP structure**

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 polarizations, say **w***<sup>i</sup>* and **w***j*, of each SM *alone, ck* **w***i*, **w***<sup>j</sup> ,* is invariant to the choice of the passage (up to a scale factor), see (40) for details. Under the three hypotheses above it follows that:

$$E\left[y\_n\left(\mathbf{w}\_i\right)y\_m^\*\left(\mathbf{w}\_j\right)\right] = \sum\_{k=1}^K c\_k\left(\mathbf{w}\_{i\prime}, \mathbf{w}\_j\right)r\_k\left(n, m\right) \forall \mathbf{w}\_{i\prime}, \mathbf{w}\_{j\prime}, n\prime m \tag{25}$$

Multi-Baseline SARs 21

Forest Structure Retrieval from Multi-Baseline SARs 47

(1 − *b*) **C**<sup>1</sup> − *b***C**<sup>2</sup>

− (1 − *a*) **C**<sup>1</sup> + *a***C**<sup>2</sup>

**R***<sup>v</sup>* = *b***R**<sup>1</sup> + (1 − *b*) **R**<sup>2</sup>

where **R***<sup>k</sup>*, **C***<sup>k</sup>* are two sets of matrices yielded by the SKP Decomposition, see (40).

It is very important to point out that the choice of the parameters (*a*, *b*) in (28) does affect the solution concerning the retrieval of the polarimetric signatures and structure matrices for ground and volume scattering, whereas the sum of their Kronecker products is invariant to choice of (*a*, *b*). In other words, the SKP Decomposition yields two sets of matrices that can be linearly combined so as to reconstruct exactly the structure matrices and polarimetric signatures associated with ground and volume scattering. Yet, the coefficients of such a linear combination are not known. An obvious criterion to eliminate non-physically valid solutions is to admit only values of (*a*, *b*) that yield, through (28), (semi)positive definite polarimetric signatures and structure matrices for both ground and volume scattering. We will define the set of values of (*a*, *b*) corresponding to physically valid solutions as Region of Physical Validity

Suppose now that a certain matrix *Wmod* is the best estimate of the true data covariance matrix in a certain metric, and also suppose that **W***mod* can be written as the sum of 2 KPs. What equations (28) state is that there exist infinite ways of representing the matrix *Wmod*, each of which corresponding to a particular value of the parameters (*a*, *b*). The parameters (*a*, *b*) then represents the ambiguity of the ground-volume decomposition problem, meaning that by varying (*a*, *b*) within the RPV we end up with different physically valid polarimetric signatures and structure matrices, yet entailing no variations of the degree of fitness of *Wmod* with respect to the original data. In other words, each *solution* of the ambiguity associated with the choice of (*a*, *b*) within the RPV represents a particular *physically valid and data-consistent model* for both the polarimetric signatures and the structure matrices of ground and volume

In this section we discuss the shape of the RPV, under the assumption that the data covariance matrix is actually a sum of *K* = 2 KPs representing ground and volume scattering, according

An exact procedure for the determination of the RPV has been derived in (40), by recasting equation (28) in diagonal form. Although the procedure is straightforward, its description involves quite lengthy matrix manipulations. For this reason, we will discuss here the shape of the RPV by resorting to a useful geometrical interpretation. To do this we will assume the Polarimetric Stationarity Condition to hold (43), resulting in the structure matrix of each SM to have unitary elements on the main diagonal, see(40). This property is inherited by the matrices **R**1,**R**<sup>2</sup> in (28), which allows to interpret the off diagonal elements of **R***g*,**R***<sup>v</sup>* as the complex interferometric coherence of ground and volume scattering (the elements on their diagonal being identically equal to 1). It is then immediate to see from equation (28) that, in each interferogram, ground and volume coherences are bound to lie on a straight line in the complex plane. Accordingly, the optimal choice of the parameters (*a*, *b*) is the

*a* − *b*

*a* − *b*

**<sup>C</sup>***<sup>g</sup>* <sup>=</sup> <sup>1</sup>

**<sup>C</sup>***<sup>v</sup>* <sup>=</sup> <sup>1</sup>

(RPV). Details about the RPV are provided in the remainder.

**R***<sup>g</sup>* = *a***R**<sup>1</sup> + (1 − *a*) **R**<sup>2</sup> (28)

such that:

scattering.

to model (27).

**4.4 Regions of Physical Validity (RPV)**

where *K* is the total number of SMs. Chosen an arbitrary polarization basis, we can recast (25) in matrix form as:

$$\mathbf{W} = \sum\_{k=1}^{K} \mathbf{C}\_{k} \otimes \mathbf{R}\_{k} \tag{26}$$

where: ⊗ denotes Kronecker product; **W** ( *N* · *Np* × *N* · *Np* ) is the covariance matrix of the multi-baseline and multi-polarimetric data; the matrices **C***<sup>k</sup>* ( *Np* × *Np* ) and **R***<sup>k</sup>* ([*N* × *N*]) are such that *ck* **w***i*, **w***<sup>j</sup>* = **w***<sup>H</sup> <sup>i</sup>* **C***k***w***<sup>j</sup>* and *rk* (*n*, *m*) = {**R***k*}*nm* ({}*nm* denoting the *nm* − *th* element of a matrix).

Accordingly, model (26) allows to represent each SM through a Kronecker product between two matrices. The first, **C***k*, accounts for the correlation among different polarizations, and thus it represents the polarimetric properties of the *k* − *th* SM, see for example (41). The second, **R***k*, accounts for the correlation among different baselines, therefore carrying the information about the vertical structure of the *<sup>k</sup>* <sup>−</sup> *th* SM 3. The matrices **<sup>C</sup>***k*, **<sup>R</sup>***<sup>k</sup>* will be hereinafter referred to as polarimetric signatures and structure matrices, respectively, by virtue of their physical meaning.

Following the arguments in (40), it can be shown that in most cases it can be assumed that the data covariance matrix is constituted by just two Kronecker Products, one associated with volume backscattering, and the other with the ensemble of all SMs whose phase center is ground locked, namely surface backscattering, ground-volume scattering, and ground-trunk scattering. Accordingly, eq. (26) defaults to:

$$\mathbf{W} = \mathbf{C}\_{\mathcal{S}} \otimes \mathbf{R}\_{\mathcal{S}} + \mathbf{C}\_{\mathcal{v}} \otimes \mathbf{R}\_{\mathcal{v}} \tag{27}$$

where the subscript *v* refers to volume backscattering and the subscript *g* refers to all SMs whose phase center is ground locked. For sake of simplicity, hereinafter we will refer to the two terms in (27) simply as ground scattering and volume scattering. It is worth noting that the assumption of two Kronecker Products can be approximately retained also in the case of an Oriented Volume over Ground (OVoG), by interpreting the matrix **R***<sup>v</sup>* as the average volume structure matrix across all polarizations (40).

#### **4.3 Model representation**

The key to the exploitation of the SKP structure is the important result, due to Van Loan and Pitsianis (42), after which *every* matrix can be decomposed into a SKP. It is shown in (40) that the terms of the SKP Decomposition are related to the matrices **C***k*, **R***<sup>k</sup>* via a linear, invertible transformation, which is defined by exactly *K*(*K* − 1) real numbers. Assuming *K* = 2 SMs, as in the case of ground and volume scattering, it follows that there exist 2 real numbers (*a*, *b*)

$$\{\mathbf{R}\_k\}\_{nm} = \int \mathbf{S}\_k\left(z\right) \exp\left\{j\left(k\_z\left(n\right) - k\_z\left(m\right)\right)z\right\} dz$$

<sup>3</sup> Neglecting temporal decorrelation and assuming target stationarity, the *nm* <sup>−</sup> *th* entry of the matrix **Rk** is obtained as (18), (6), (26):

where *z* is the vertical coordinate, *Sk* (*z*) is the vertical profile of the backscattered power for the *k* − *th* SM, *kz* (*n*) is the height to phase conversion factor for the *n* − *th* image (7). It is worth noting that eventual temporal coherence losses would be completely absorbed by the matrices **R***k*, as discussed in (40). Accordingly, in presence of temporal decorrelation nothing changes as for the validity of model (26), but it should be kept in mind that in this case the matrices **R***<sup>k</sup>* would represent not only the spatial structure, but also the temporal behavior of the *k* − *th* SM.

such that:

20 Will-be-set-by-IN-TECH

where *K* is the total number of SMs. Chosen an arbitrary polarization basis, we can recast (25)

Accordingly, model (26) allows to represent each SM through a Kronecker product between two matrices. The first, **C***k*, accounts for the correlation among different polarizations, and thus it represents the polarimetric properties of the *k* − *th* SM, see for example (41). The second, **R***k*, accounts for the correlation among different baselines, therefore carrying the information about the vertical structure of the *<sup>k</sup>* <sup>−</sup> *th* SM 3. The matrices **<sup>C</sup>***k*, **<sup>R</sup>***<sup>k</sup>* will be hereinafter referred to as polarimetric signatures and structure matrices, respectively, by

Following the arguments in (40), it can be shown that in most cases it can be assumed that the data covariance matrix is constituted by just two Kronecker Products, one associated with volume backscattering, and the other with the ensemble of all SMs whose phase center is ground locked, namely surface backscattering, ground-volume scattering, and ground-trunk

where the subscript *v* refers to volume backscattering and the subscript *g* refers to all SMs whose phase center is ground locked. For sake of simplicity, hereinafter we will refer to the two terms in (27) simply as ground scattering and volume scattering. It is worth noting that the assumption of two Kronecker Products can be approximately retained also in the case of an Oriented Volume over Ground (OVoG), by interpreting the matrix **R***<sup>v</sup>* as the average

The key to the exploitation of the SKP structure is the important result, due to Van Loan and Pitsianis (42), after which *every* matrix can be decomposed into a SKP. It is shown in (40) that the terms of the SKP Decomposition are related to the matrices **C***k*, **R***<sup>k</sup>* via a linear, invertible transformation, which is defined by exactly *K*(*K* − 1) real numbers. Assuming *K* = 2 SMs, as in the case of ground and volume scattering, it follows that there exist 2 real numbers (*a*, *b*)

<sup>3</sup> Neglecting temporal decorrelation and assuming target stationarity, the *nm* <sup>−</sup> *th* entry of the matrix **Rk**

where *z* is the vertical coordinate, *Sk* (*z*) is the vertical profile of the backscattered power for the *k* − *th* SM, *kz* (*n*) is the height to phase conversion factor for the *n* − *th* image (7). It is worth noting that eventual temporal coherence losses would be completely absorbed by the matrices **R***k*, as discussed in (40). Accordingly, in presence of temporal decorrelation nothing changes as for the validity of model (26), but it should be kept in mind that in this case the matrices **R***<sup>k</sup>* would represent not only the spatial

*Sk* (*z*)*exp* {*j*(*kz* (*n*) − *kz* (*m*)) *z*} *dz*

*N* · *Np* × *N* · *Np*

*<sup>i</sup>* **C***k***w***<sup>j</sup>* and *rk* (*n*, *m*) = {**R***k*}*nm* ({}*nm* denoting the *nm* − *th* element

*Np* × *Np*

**W** = **C***<sup>g</sup>* ⊗ **R***<sup>g</sup>* + **C***<sup>v</sup>* ⊗ **R***<sup>v</sup>* (27)

**C***<sup>k</sup>* ⊗ **R***<sup>k</sup>* (26)

) is the covariance matrix of the

) and **R***<sup>k</sup>* ([*N* × *N*]) are

*K* ∑ *k*=1

**W** =

in matrix form as:

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

virtue of their physical meaning.

**4.3 Model representation**

is obtained as (18), (6), (26):

scattering. Accordingly, eq. (26) defaults to:

volume structure matrix across all polarizations (40).

{**R***k*}*nm* =

structure, but also the temporal behavior of the *k* − *th* SM.

ˆ

such that *ck*

of a matrix).

where: ⊗ denotes Kronecker product; **W** (

multi-baseline and multi-polarimetric data; the matrices **C***<sup>k</sup>* (

$$\begin{aligned} \mathbf{R}\_{\mathcal{S}} &= a\widetilde{\mathbf{R}}\_1 + (1 - a)\,\widetilde{\mathbf{R}}\_2 \\ \mathbf{R}\_v &= b\widetilde{\mathbf{R}}\_1 + (1 - b)\,\widetilde{\mathbf{R}}\_2 \\ \mathbf{C}\_{\mathcal{S}} &= \frac{1}{a - b} \left( (1 - b)\,\widetilde{\mathbf{C}}\_1 - b\widetilde{\mathbf{C}}\_2 \right) \\ \mathbf{C}\_v &= \frac{1}{a - b} \left( -(1 - a)\,\widetilde{\mathbf{C}}\_1 + a\widetilde{\mathbf{C}}\_2 \right) \end{aligned} \tag{28}$$

where **R***<sup>k</sup>*, **C***<sup>k</sup>* are two sets of matrices yielded by the SKP Decomposition, see (40).

It is very important to point out that the choice of the parameters (*a*, *b*) in (28) does affect the solution concerning the retrieval of the polarimetric signatures and structure matrices for ground and volume scattering, whereas the sum of their Kronecker products is invariant to choice of (*a*, *b*). In other words, the SKP Decomposition yields two sets of matrices that can be linearly combined so as to reconstruct exactly the structure matrices and polarimetric signatures associated with ground and volume scattering. Yet, the coefficients of such a linear combination are not known. An obvious criterion to eliminate non-physically valid solutions is to admit only values of (*a*, *b*) that yield, through (28), (semi)positive definite polarimetric signatures and structure matrices for both ground and volume scattering. We will define the set of values of (*a*, *b*) corresponding to physically valid solutions as Region of Physical Validity (RPV). Details about the RPV are provided in the remainder.

Suppose now that a certain matrix *Wmod* is the best estimate of the true data covariance matrix in a certain metric, and also suppose that **W***mod* can be written as the sum of 2 KPs. What equations (28) state is that there exist infinite ways of representing the matrix *Wmod*, each of which corresponding to a particular value of the parameters (*a*, *b*). The parameters (*a*, *b*) then represents the ambiguity of the ground-volume decomposition problem, meaning that by varying (*a*, *b*) within the RPV we end up with different physically valid polarimetric signatures and structure matrices, yet entailing no variations of the degree of fitness of *Wmod* with respect to the original data. In other words, each *solution* of the ambiguity associated with the choice of (*a*, *b*) within the RPV represents a particular *physically valid and data-consistent model* for both the polarimetric signatures and the structure matrices of ground and volume scattering.

#### **4.4 Regions of Physical Validity (RPV)**

In this section we discuss the shape of the RPV, under the assumption that the data covariance matrix is actually a sum of *K* = 2 KPs representing ground and volume scattering, according to model (27).

An exact procedure for the determination of the RPV has been derived in (40), by recasting equation (28) in diagonal form. Although the procedure is straightforward, its description involves quite lengthy matrix manipulations. For this reason, we will discuss here the shape of the RPV by resorting to a useful geometrical interpretation. To do this we will assume the Polarimetric Stationarity Condition to hold (43), resulting in the structure matrix of each SM to have unitary elements on the main diagonal, see(40). This property is inherited by the matrices **R**1,**R**<sup>2</sup> in (28), which allows to interpret the off diagonal elements of **R***g*,**R***<sup>v</sup>* as the complex interferometric coherence of ground and volume scattering (the elements on their diagonal being identically equal to 1). It is then immediate to see from equation (28) that, in each interferogram, ground and volume coherences are bound to lie on a straight line in the complex plane. Accordingly, the optimal choice of the parameters (*a*, *b*) is the

(29)

Multi-Baseline SARs 23

Forest Structure Retrieval from Multi-Baseline SARs 49

Scene Semi-boreal forest

Vertical resolution 10 m (near range) to 40 m (far range)

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

• The inner boundary solution on branch *a* results in the volume polarimetric signature to be rank-deficient. This entails the existence of a polarization where volume scattering does not contribute, after which it follows that this solution is not consistent with physical model for forest scattering (44), (38). The ground structure matrix is characterized by the lowest coherence values, being contaminated by volume contributions. Accordingly, such

• The outer boundary solution on branch *a* results in the volume polarimetric signature to be full rank, consistently with physical model for forest scattering. The resulting ground structure matrix is characterized by the highest coherence values compatible with the RPV. Provided the number of tracks is sufficient, this solution yields an unbiased estimation of

• The inner boundary solution on branch *b* results in the ground polarimetric signature to be rank-deficient, consistently with the hypothesis that there exists one polarization where volume only contributions are present. If this is true, the resulting volume structure corresponds to the true one. Otherwise, the result is systematically contaminated by ground contributions, resulting in apparent volume contributions close to the ground. • The outer boundary solution on branch *b* results in the ground polarimetric signature to be full rank. Accordingly, this solution accounts for the presence of ground-locked contributions in all polarizations. The resulting volume structure matrix is maximally coherent. If few tracks are employed, this solution acts as an high-pass filter, resulting in the volume to appear thinner than it is. As the number of available baselines increases, this solution converges to the true structure for volume contributions in the upper vegetation layers, whereas volume contributions from the ground level are absorbed into the ground

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 ground coherences even in presence of coherence losses.

Site Remningstorp, Central Sweden

Campaign BioSAR 2007 Acquisition System E-SAR - DLR Acquisition Period Spring 2007

Topography Flat Tomographic Tracks 9 - Fully Polarimetric Band P-Band

Slant Range resolution 2 m Azimuth resolution 1.6 m

Table 2. The BIOSAR 2007 data-set

a solution is to be discarded.

**4.5 Physical interpretation**

boundaries of the RPV:

structure.

**4.6 Case studies**

the analyzed data are reported in tables 2, 3, 4.


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 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 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 varying *a* and *b*, respectively.

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 progressive elimination of incorrect solutions.

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


Table 2. The BIOSAR 2007 data-set
