*3.3. Assessment of Antenna Cross-Talks Using an Iterative Method*

Van Zyl's Equations in Terms of *RR* for Azimuthally Symmetric Target

Under the assumption of an azimuthally symmetric target, the following equations are derived from [16] as a function of RR and the cross-talk estimate ∆<sup>1</sup> and ∆ 0 2 [6]:

$$ \simeq \quad \Delta\_1 < |V\_{hh}|^2 > + \Delta\_2' \cdot < V\_{vv}V\_{hh}^\* > + \tag{7}$$

$$2\Delta\_2'^\* \cdot < |V\_{hw}'|^2 > \cdot (1 - \text{RR})$$

$$ \simeq \begin{aligned} \Delta\_1 \cdot &< V\_{hh}V\_{vv}^\* > + \Delta\_2' \\ &< |V\_{vv}|^2 > + 2\Delta\_1^\* \cdot < |V\_{hv}'|^2 > \cdot (1 - \text{RR}) \end{aligned} \tag{8}$$

The two unknowns ∆<sup>1</sup> and ∆ 0 2 can be determined by solving the two equations above as a function of the voltages *Vhh*, *V* 0 *hv*, *Vvv*, and their cross-correlations measured over the azimuthally symmetrical reference target [6]. The two unknowns are determined using an analytical equation or an iterative method depending on the *RR* values:


The iterative method is applied as follows:


After the estimation of ∆<sup>1</sup> and ∆ 0 2 , all the required calibration parameters ∆1, ∆2, *F*1, and *F*<sup>2</sup> can be deduced using the additional equations provided by the corner reflector, as shown in [16].

In the following, the iterative method described above is adopted for the assessment of PALSAR2 system parameters using datasets collected over the Amazonian rainforest sites (with CRs), at free Faraday rotation conditions.

## **4. Assessment of PALSAR2 System Parameters at Low Faraday Rotation Conditions Using Amazonian Rainforests**

#### *4.1. Calibration Sites*

The Amazonian rainforest near the geomagnetic equator, at free Faraday rotation [15], has been used as the ideal site for the assessment and calibration of L-band ALOS-PALSAR and PALSAR2 [3,5,6,22]. CRs deployed at free Faraday rotation provide accurate estimation of antenna cross-talks in contrast to the biased measurements obtained with CRs deployed at significant Faraday rotation, as shown in [6]. In 2014, five CRs were deployed by JAXA in the Amazonian rainforests [23], and used in support of polarimetric PALSAR2 calibration. Six polarimetric images collected over the Amazonian rainforests (near Rio Branco, Brazil) at various incidence angles are used in the following to assess PALSAR-2 distortion matrix for five beams (FP6-3 to FP6-7) with incidence angle varying from 25◦ to 40◦ . The CR measurements are also used to confirm the very low Faraday rotation conditions during PALSAR2 data acquisition.

#### *4.2. Assessment of Polarimetric PALSAR2 Distortion Matrix*

Figures 1–4 present the response (in range) of a CR covered with the FP6-3 polarimetric mode on the 8th of August, 2014. According to Figures 3 and 4, there is no CR (HV and VH) return at the HH and VV peak location. This shows that the PALSAR2 cross-talk is very low, and the ALOS2 acquisition took place at very low Faraday rotation conditions. The various PALSAR2 modes considered herein, the dates of the acquisition, and the incidence angle (in degree) at the CR location are indicated in Table 1. Channel imbalance (*F*<sup>1</sup> and *F*2) (magnitude and phase) are also given in Table 1. The following observations can be noted:


The extended Freeman–van Zyl iterative method is applied for the generation of PALSAR2 antenna cross-talks. ∆<sup>1</sup> and ∆<sup>2</sup> are given (in dB) in Table 2. The residual error *RR* is computed using Equation (5). The final error *RR f* computed at the last iteration corresponds to the residual error. All the cross-talks given in the table were obtained with a very low residual error (*RR f* lower than −43 dB).

**Mode Date CR Inc** |*f***1**| *φ<sup>f</sup>* **<sup>1</sup>** |*f***2**| *φ<sup>f</sup>* **<sup>2</sup>** FP6-3 8−08 28 1.06 −1.1 1.26 −28.0 FP6-4 8−22 31 1.06 −22.1 1.01 −25.6 FP6-5 9−05 35 1.03 −5.4 1.01 −28.2 FP6-6 9−19 37 1.06 −23.6 1.04 −27.4 FP6-7 8−13 39 1.05 −3.3 1.02 −25.7 FP6-7 8−27 39 1.04 −3.7 1.01 −25.7

**Table 1.** PALSAR2 channel imbalance (magnitude and phase (in degree)).

**Table 2.** PALSAR2 Antenna Cross-Talks (in dB) and Faraday Rotation (in degree).


**Figure 1.** Corner reflector reflector HH response (in range expressed in pixel numbers).

**Figure 2.** Corner reflector VV response (in range).

**Figure 3.** Corner reflector HV response (in range).

**Figure 4.** Corner reflector VH response (in range).

As can be noted in Table 2, the PALSAR2 antenna is highly isolated with cross-talk lower than −40 dB. The CR-measurements permit demonstrating the very high isolation of the PALSAR2 antenna: cross-talks lower than −40 dB [24]. Similar results were obtained in [3,22,25] using different calibration approaches.

#### *4.3. Measurement of the Actual Faraday Rotation during PALSAR2 Acquisition*

Many studies have shown that Faraday rotation is low near the equator [15,26]. These results are confirmed in the following for the five PALSAR2 images collected over the JAXA sites of calibration in the Amazonian rainforests. Several methods can be used to measure the Faraday rotation [26–28]. We have adopted herein the Bickel and Bates method [29] which calculates the Faraday rotation at the circular basis as follows:

$$
\begin{bmatrix} Z\_{11} & Z\_{12} \\ Z\_{21} & Z\_{22} \end{bmatrix}^T = \begin{bmatrix} 1 & j \\ j & 1 \end{bmatrix}^T [\mathbf{S}] \boldsymbol{\Omega} \begin{bmatrix} 1 & j \\ j & 1 \end{bmatrix} \tag{9}
$$

where [*S*]<sup>Ω</sup> is given as a function of the Faraday rotation angle Ω and the scattering matrix [S] of Equation (3) by:

$$[\mathbb{S}]\_{\Omega} = \begin{bmatrix} \cos \Omega & -\sin \Omega \\ \sin \Omega & \cos \Omega \end{bmatrix} [\mathbb{S}] \begin{bmatrix} \cos \Omega & -\sin \Omega \\ \sin \Omega & \cos \Omega \end{bmatrix} \tag{10}$$

Ω is given by [27,29]:

$$
\Omega = <\frac{1}{4} \cdot \arg(Z\_{12} Z\_{21}^\*)> \tag{11}
$$

where *arg*(*Z*) is the argument of the complex *Z*, and <> denotes spatial averaging.

Equation (11) is used to measure the Faraday rotation for the five PALSAR images. The measurements obtained at the CRs are given in Table 2. As expected, the Faraday rotation angle is very low (lower than 0.24◦ ) during the 5 PALSAR2 acquisitions. This confirms the accuracy of the PALSAR2 antenna distortion matrix measurements obtained above, under the assumption of negligible Faraday rotation.

#### **5. Impact of Significant Faraday Rotation on PALSAR2 Polarimetric Distortion Matrix Measurement**

The model of Equation (1) can be extended to take into account the Faraday rotation. Channel imbalances can also be separated from the polarimetric distortion matrices as done in [21]. This leads to the following equation [21,30]:

$$[V] = \begin{bmatrix} 1 & 0 \\ 0 & F\_1 \end{bmatrix} \begin{bmatrix} 1 & \delta\_2 \\ \delta\_1' & 1 \end{bmatrix} [\mathbf{S}]\_\Omega \begin{bmatrix} 1 & \delta\_3' \\ \delta\_4 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & F\_2 \end{bmatrix} \tag{12}$$

with *δ* 0 <sup>1</sup> = *δ*1/*F*1, *δ* 0 <sup>3</sup> = *δ*3/*F*2, and [*S*]<sup>Ω</sup> is given by Equation (10) as a function of the scattering matrix [*S*] and the Faraday rotation angle Ω. It is worth noting that Equation (12), which is an extension of the van Zyl calibration model that takes into account the Faraday rotation, is expressed as a function of the conventional scattering matrix [*S*] in contrast to the Freeman model [30] expressed as a function of the transposes of [*S*] and [*V*] matrices.

The CR response combined with the channel imbalance phase difference, which can be derived using the Zebker method [31], permits the measurement of the channel imbalance *F*<sup>1</sup> and *F*<sup>2</sup> ([6,17,31]). After the channel imbalance correction, the measured voltages can be expressed (using Equation (12)) as a function of the antenna cross-talks and [*S*]<sup>Ω</sup> matrix elements in the following equation:

$$
\begin{bmatrix} V\_{hh} \\ V\_{hv} \\ V\_{vh} \\ V\_{vv} \end{bmatrix}\_{\begin{bmatrix} \delta\_{hh}^{\Omega} + \delta\_{4}\mathcal{S}\_{h\overline{v}}^{\Omega} + \delta\_{2}\mathcal{S}\_{v\overline{h}}^{\Omega} \\ \delta\_{3}'\mathcal{S}\_{h\overline{h}}^{\Omega} + \mathcal{S}\_{h\overline{v}}^{\Omega} + \delta\_{2}\mathcal{S}\_{v\overline{v}}^{\Omega} \\ \delta\_{1}'\mathcal{S}\_{h\overline{h}}^{\Omega} + \mathcal{S}\_{hv}^{\Omega} + \delta\_{4}\mathcal{S}\_{v\overline{v}}^{\Omega} \\ \delta\_{1}'\mathcal{S}\_{h\overline{v}}^{\Omega} + \delta\_{3}'\mathcal{S}\_{v\overline{h}}^{\Omega} + \mathcal{S}\_{v\overline{v}}^{\Omega} \end{bmatrix} \tag{13}
$$

where

$$
\begin{bmatrix}
\mathcal{S}\_{hh}^{\Omega} \\
\mathcal{S}\_{hv}^{\Omega} \\
\mathcal{S}\_{vh}^{\Omega} \\
\mathcal{S}\_{vv}^{\Omega}
\end{bmatrix} = \begin{bmatrix}
\cos^2\Omega \cdot \mathcal{S}\_{hh} - \sin^2\Omega \cdot \mathcal{S}\_{vv} \\
\mathcal{S}\_{hv} - \sin\Omega\cos\Omega \cdot (\mathcal{S}\_{hh} + \mathcal{S}\_{vv}) \\
\mathcal{S}\_{hv} + \sin\Omega\cos\Omega \cdot (\mathcal{S}\_{hh} + \mathcal{S}\_{vv}) \\
\end{bmatrix} \tag{14}
$$

For a CR, *Shh* = *Svv* = *K* and *Shv* = *Svh* = 0, and Equation (13) can be used to derive the CR-measured voltage vector as a function of the Faraday rotation angle Ω and the antenna cross-talks (*δ* 0 1 , *δ*2, *δ* 0 3 , *δ*4):

$$
\begin{bmatrix} V\_{hh}^{\text{CR}} \\ V\_{h\nu}^{\text{CR}} \\ V\_{v\nu}^{\text{CR}} \\ V\_{vv}^{\text{CR}} \end{bmatrix} = -K \begin{bmatrix} \cos 2\Omega + \sin 2\Omega (\delta\_2 - \delta\_4) \\ \cos 2\Omega (\delta\_2 + \delta\_3') - \sin 2\Omega \\ \cos 2\Omega (\delta\_1' + \delta\_4) + \sin 2\Omega \\ \cos 2\Omega - \sin 2\Omega (\delta\_1' - \delta\_3') \end{bmatrix} \tag{15}
$$

In 2014, several PALSAR2 images were collected over the CCRS calibration site in Ottawa. The PALSAR2 image collected on the 9th of September is used herein. Figures 5–8 present the polarimetric response of the CCRS (2.5 m CR) at HH, VV, HV, and VH. In contrast to the low return at cross-pol of the JAXA CR deployed in the Amazonian rainforests (Figures 3 and 4), the Ottawa CR presents a significant return at HV and VH polarization as seen in Figures 7 and 8.

**Figure 5.** Corner reflector HH response (in range expressed in pixel numbers).

**Figure 6.** Corner reflector VV response (Ottawa).

**Figure 7.** Corner reflector HV response (Ottawa).

**Figure 8.** Corner reflector VH response (Ottawa).

The results obtained in the previous section show that the PALSAR2 antenna is highly isolated with cross-talks lower than −40 dB. The analysis of the CR voltage Equation (15) with negligible cross-talks (cross-talks lower than −40 dB) leads to the following conclusions:

1. The ratios of the cross-like polarization voltage, *V CR hv* /*V CR hh* = −20.03 dB and *V CR vh* /*V CR vv* =−19.95 dB do not correspond to the actual antenna cross-talk contamination of the cross-pol (HV and VH) by the like polarization (HH and VV), as could be misinterpreted if the Faraday rotation contamination is ignored ([7]). In fact, these cross-like polarization voltage ratios lead to an estimation of the Faraday rotation angle Ω for a highly isolated antenna, as can be shown using Equation (15):

$$
\begin{bmatrix}
\frac{V\_{\text{lv}}^{\text{CR}}}{V\_{\text{v}\text{R}}^{\text{CR}}}\\\frac{V\_{\text{v}\text{R}}^{\text{CR}}}{V\_{\text{vv}}^{\text{CR}}}
\end{bmatrix}
\quad = \quad K \cdot \begin{bmatrix}
\end{bmatrix}
\tag{16}
$$

Equation (16) is used to estimate the Faraday rotation during PALSAR2 acquisitions. The results obtained are similar (within 0.2) with the ones obtained with the Bickel and Bates method (Equation (11)); Ω = 2.8◦ with *V CR hv* /*V CR hh* , Ω = 2.9◦ with *V CR vh* /*V CR vv* in comparison with Ω = 3.1◦ obtained over a forested area using Equation (11).

2. The sum of the CR cross-pol voltages cancels the Faraday rotation contamination.

  This can be shown using the following equation derived from (15):

$$((V\_{hv}^{\mathbb{C}R} + V\_{vh}^{\mathbb{C}R}) / \mathcal{K} = (\delta\_1' + \delta\_2 + \delta\_3' + \delta\_4) \cos 2\Omega \tag{17}$$

Since the cross-talks are negligible, (*V CR hv* + *V CR vh* ) should be close to zero. Figure 9 presents the CR response of the averaged PALSAR2 image (*hv* + *vh*)/2. The significant peaks that occur at HV and VH images of Figures 6 and 7 vanish in Figure 9. At the HH and VV peak location, the intensity of the averaged cross-pol return vanishes in the CR surrounding clutter of radar backscattering (−18.23 dB) which is much lower than HH and VV retro-diffusion (about 15 dB).

In summary, the analytical presentation of the CR voltage as a function of the antenna cross-talks and Faraday rotation leads to the conclusion that the HV and VH contaminations, which can be measured from SAR images collected with a highly isolated antenna, are mainly due to the actual Faraday rotation. In the case that the antenna is not highly isolated, Equation (13) obtained after the channel imbalance correction (using the CR) shows that additional reference point targets (of scattering response different from the CR) should be used for the measurement of the 5 unknowns; the 4 antenna cross-talks and the Faraday rotation.

**Figure 9.** Corner reflector (HV + VH)/2 response (Ottawa).

#### **6. Conclusions**

In summary, the PALSAR antenna is highly isolated. Both the antenna subarray and T/R are highly isolated, and as a result, the global antenna is highly isolated for all the modes considered. The excellent performances of the polarimetric PALSAR2 in terms of NESZ (better than −37 dB ([1,3]) combined with the high antenna isolation permit the demonstration of the unique L-band long-penetration SAR capabilities at low incidence angles (25◦ ) for subsurface peatland hydrology monitoring and discontinuous permafrost mapping [32,33]. The high isolation of the polarimetric ALOS2-PALSAR2 permits a simplification of the assessment and calibration of the ALOS2-Compact experimental mode, as discussed in [34].

**Author Contributions:** Concepts and Methodology, R.T. and M.S.; ALOS2-PALSAR2 raw data processing and Calibration, M.S. and T.M.; writing—original draft preparation, R.T. and M.S.; writing—review and editing, R.T., M.S. and T.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** The ALOS-2 original data are provided by JAXA under ALOS-2 Calibration Validation and Science Team (CVST) and the Research Announcement on the Earth Observations (EO-RA).

**Acknowledgments:** The authors would like to thank the anonymous reviewers for their helpful comments and suggestions. We would like to express our special thanks to JAXA for data observation and provisions. S. Nedelcu from CCRS is thanked for his help in PALSAR2 data processing.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

