DRDC-RDDC-2016-N032

COMPACT POLARIMETRIC SAR PRODUCT AND CALIBRATION CONSIDERATIONS FOR TARGET ANALYSIS

Ramin Sabry

Space and ISR Applications, DRDC Ottawa Research Centre, Ottawa, Ontario,

ABSTRACT

Compact polarimetric (CP) data exploitation is currently of growing interest considering the new generation of such Synthetic Aperture Radar (SAR) systems. These systems offer target detection and classification capabilities comparable to those of polarimetric SARs (PolSAR) with less stringent requirements. A good example is the Constellation Mission (RCM). In this paper, some characteristic CP products are described and effects of CP mode deviation from ideal circular polarization transmit on classifications are modeled. The latter is important for operation of typical CP modes (e.g., RCM). The developed model can be used to estimate the ellipticity variation from CP measured data, and hence, calibrate the classification products.

Keywords: Synthetic Aperture Radar (SAR), Compact Polarimetry (CP), CP Calibration

1. INTRODUCTION

Compact polarimetric (CP) Synthetic Aperture Radar (SAR) systems have become of growing interest as a step-up from single-pol systems toward full polarimetry1-4. Such systems provide more target information than single-channel SAR, even comparable to polarimetric SAR for certain applications, with less stringent data and energy budget requirements. C-band CP data across a variety of swath widths and resolutions will be operationally available from RADARSAT Constellation Mission (RCM). One notes that significant information available in fully polarimetric data for target analysis is not offered by CP data (as implied by compact nomenclature). However, valuable target information may be extracted2-3 and optimized through proper exploitation of CP data within a unified polarimetric framework5.

One important consideration for effective CP data exploitation is the uncertainty associated with the actual transmit waveform (e.g., propagation channel, transmitter and antenna performance) that results in non-ideal incident wave ellipticity. This becomes more important for data exploitation of circular transmit CP modes, e.g., dual circular polarization (DCP), circular-transmit/linear-receive (CTLR), which represent typical CP SAR system modes6 (e.g.,

© Her Majesty the Queen in Right of Canada, as represented by the Minister of National Defence, 2016 © Sa Majesté la Reine (en droit du Canada), telle que représentée par le ministre de la Défense nationale, 2016 RCM) due to their backscattering advantages2,5 . Thus, it is useful to have an accurate estimate of such circular polarization imperfections and their effect on CP exploitation products. Moreover, the developed model can be used for CP data calibration. This work intends to analytically model and quantify effects of the described non-circular polarization.

To satisfy the stated objective, the problem in hand can be succinctly expressed as modeling scattering observables (e.g., scattering mechanisms) variation in terms of ellipticity deviation from perfect circular polarization. In other words, the overall effect of all contributing factors can be modeled by an ellipticity variance associated with the polarized portion of incident wave. Estimating such variance introduces a challenge by itself. As will be discussed, however, variation of classification indicators for a range of ellipticity variation that is typical (expected) for CP applications can be analytically modeled. Hence, efficiency of a CP approach can be evaluated for applications of interest. Analytic description of the core CP observable/product, i.e., scattered Stokes vector, that generates a variety of classification features will be derived for a general ellipticity transmit waveform in the vicinity of circular polarization operation point. Subsequently, variations of the classification features with respect to the assumed/ideal ellipticity (i.e., circular) will identify error and be used to aid potential calibration.

2. ANALYTIC DESCRIPTION OF THE SCATTERED FIELD

Scattered field Stokes vector is related to that of the incident field by the Mueller matrix. For general bistatic applications, the 4×4 Mueller matrix [M] can be expressed in terms of the conventional 4×4 coherency scattering matrix [T] through a 1:1 mapping. The coherency matrix [T] contains target information and is independent of the incident or transmit field. Therefore, one can exploit the observed or measured Stokes vector of the scattered field to extract certain coherency matrix [T] elements which include valuable target scattering information (e.g., odd or even bounce, volume, surface and surface roughness) for a known incident waveform. It is evident (as hinted) that all coherency matrix elements cannot be derived considering the number of unknowns (i.e., [T] matrix) and number of knowns or measurements (i.e., Stokes Vector) and, in general, a priori assumptions/models are needed.

For an ideal circularly polarized target illumination, the incident wave Stokes vector is given by:

(0) ⎡⎤Ginc ⎡⎤1 ⎢⎥(1) ⎢⎥ G 0 GL==⎢⎥inc ⎢⎥ (:HC+sign ,RHC ≡−sign ) inc ⎢⎥(2) Ginc ⎢⎥0 ⎢⎥(3) ⎢⎥ ⎣⎦⎢⎥Ginc ⎣⎦±1 (1)

As indicated, (+) and ( ̶ ) signs denote left (LHC) and right (RHC) circular transmit.

Using Mueller matrix (expressed in terms of coherency matrix) and assuming reflection symmetry7, the scattered field Stokes vector becomes:

(0) ⎡⎤Grec ⎡⎤TTT11++ 22 33 ⎢⎥⎢⎥ G (1) 2Re()T GT==⎢⎥rec ⎢⎥12 []==T;i ,j 1..4 rec ⎢⎥(2) ⎢⎥()ij Grec m 2Im()T12 ⎢⎥⎢⎥ (3) ±−TTT + + ⎣⎦⎢⎥Grec ⎣⎦⎢⎥()11 22 33 (2)

It is clear from (2) that the scattered Stokes vector elements for a LHC/RHC incident wave produce classification indicators associated with the backscattering mechanisms. Depending on the choice of incident LHC or RHC,

(0) (3) GGrec± rec produce scattering indicators T11 (single-bounce) and TT22+ 33 (double-bounce plus volume).

(0) (3) (0) (3) Furthermore, for a surface scattering scenario (i.e.,T33 →0 ), GGrec+ rec and GGrec− rec (again depending on the choice of LHC or RHC) are clear indicators of single and double bounce2,5. For an imperfect LHC (or RHC) transmit, i.e., deviation from circular polarization, description of the incident field Stokes vector (expanding about the circular polarization point χψ==45oo , 0 ) transforms to:

⎡⎤1 ⎡⎤1 ⎢⎥ ⎢⎥0 ±Δsin()() 2χψ cos 2 Δ G =⇒⎢⎥ ⎢⎥ inc ⎢⎥0 ⎢⎥±Δsin()() 2χψ sin 2 Δ ⎢⎥ ⎢⎥ ⎣⎦±1 ⎣⎦⎢⎥±Δcos() 2 χ (3)

where Δχ and Δψ represent the ellipticity and roll angles deviation or rotation from circular, respectively.

Introduction of Δψ is required for unambiguous definition of a non-circular and elliptically polarized wave.

Accordingly, one can derive modified Stokes vector of the scattered field as:

⎡++±ΔΔ⎤TTT11 22 33 2Re() T12 sin ( 2χψ ) cos( 2 ) ⎢⎥ 2Re()(TTTT±+− )sin ( 2 Δχψ ) cos ( 2 Δ ) G = ⎢⎥12 11 22 33 rec ⎢⎥ ±−+(TTT11 22 33 )()()sin 2 Δχψ sin 2 Δm 2Im ()()T12 cos 2Δχ ⎢⎥ ⎣⎦⎢⎥m 2Im()()()(TT12 sin 2ΔΔ±−++Δχψ sin 2 11T 22T 33 )()cos 2 χ (4)

The Stokes vector description (4) provides a clear indication of non-circular effects on a CP-based classification scheme, and means to estimate variations of the classification indicators.

A variety of classification products can be extracted from the Stokes vector

5 (0)(1)(2)(3) components ()GGGGrec,,, rec rec rec . Define the ratio:

(0) (3) GGrec+ rec r = (0) (3) GGrec− rec (5)

For an ideal LHC transmit (using (2) and (5)):

TT22+ 33 rC = T11 (6) can be used for double and single bounce classification (high and low values) where surface scattering is dominant. For a non-ideal LHC (or RHC), (5) cannot clearly describe the double to single bounce ratio. Using (4) in (5), one obtains:

2 Re()T12 rC +Δ+tan()χχ 2 tan () Δ T11 rΔ = 2 Re()T12 1+Δ+rC tan()χχ 2 tan () Δ T11 (7) where ‘Re’ denotes the real part. In derivation of (7), it is assumed that roll angle shift Δψ is negligible or can be calibrated. Nevertheless, the definition in (7) can be easily generalized for non-zero Δψ by using (4). Using (6)-(7), the classification error generated by the incident wave non-circular anomaly can be expressed as:

rr− Error ()Δ=χ Δ C (8) rC

It is clear that for perfect circular transmit, rrΔ → C and Error → 0 . Further simplifications can be made by inspecting (7). On can show:

22 Re()TSS12 =−hh vv (9)

where Shh and Svv represent the co-pol scattering (horizontal and vertical), and denotes the standard ensemble averaging required for coherence computation. For applications involving a dominant surface scatterer (single or Re()T double bounce), the ratio 12 is negligible and can be dropped. For general error analysis, however, this ratio T11 needs to be examined for applications of interest. Statistical analysis of (7) yields an estimate of CP-based classification error for different classes of applications. It is also evident that an estimate of helicity variance Δχ can

be made by using rΔ obtained from the scattered Stokes vector and the ground truth, i.e., associated [T ] elements. Such estimation is important for CP data calibration.

3. POLSAR IMAGE BASED ANALYSIS

Various compact polarimetric applications can be explored by utilizing PolSAR imagery and generating associated CP products. The data set used here (Figure 1) is a RADARSAT-2 (C-band) fully polarimetric (fine quad-pol, FQ7) complex image with 5.2m×7.6m nominal resolution and incidence angle 25.8° (Near), 27.6° (Far). The scene depicts Vancouver (BC, CAN) on January 11, 2009. As can be seen, a variety of backscattering phenomena associated with a complex environment (e.g., marine, agriculture, man-made structures, and terrain) are present in the scene. Figure (2) depicts the Pauli decomposition that essentially represents TTT11,, 22 33 distribution. Following discussions in the

Re(T12 ) previous section, histograms of rC and are presented in Figure 4 for a number of arbitrary image chips T11 (shown in Figure 3).

The histograms in Figure 4 indicate a general trend toward a characteristic or stationary point for rC and the ratio Re()T Re(T ) 12 . To estimate the upper bound of single/double bounce classification error, the maximum value for 12 T11 T11 is assumed in (7) and classification error (8) is calculated for a range of circular anomaly (i.e., Δ=χ 2..20oo) and included in Figure 5. Inspection of Figure 5 shows that even for low degrees of circular polarization anomaly, classification error can be considerable for single-bounce dominant scattering. This error, however, is less sensitive to ellipticity variations for stronger double-bounce scattering (higher rC ). It is conceivable that for target applications with characteristic (stationary) backscattering behavior similar to the abovementioned scenario, an acceptable estimate of Δχ ( tan ()Δχ , in particular) be derived by using an asymptotic expansion of (7). Considering the expected range of Δχ variations (e.g., Δ=χ 2..20oo, 0<Δ< tan2 ( χ ) 0.13 ), such approximation near the stationary point can be shown to be effective.

Figure 1. RADARSAT-2 (C-band) quad polarimetric (fine quad-pol, FQ7) image of Vancouver, BC, CAN (January 11, 2009). Nominal resolution is 5.2mm× 7.6 ( range× azimuth ) (RGB: R=HH, G=VV, B=HV).

Figure 2. Pauli basis image of Vancouver RADARSAT-2 data set (January11, 2009), RGB: R(double-bounce) = 2 1/2 2 1/2 2 1/2 SSHH− VV , G(volume) = SHV , B(single-bounce) = SSHH+ VV

Figure 3. Image chips selected from the quad-pol image in Figure 1 for the backscattering ratio analysis

Re()T12 Figure 4. Histograms of rC and associated with the selected image chips (CH1-CH5 of Figure 3), and T11 the entire image (Whole)

Re()T Figure 5. Classification error (%) plots for circular anomaly ( Δ=χ 2..20oo) assuming 12 statistical mean T11 (upper bound) from histograms in Figure 4

4. SUMMARY

Application of SAR compact polarimetry for target backscattering data exploitation and associated considerations are briefly explored in this work. Analytic description of the scattered field Stokes vector is derived in terms of target scattering coherency matrix, and transmit or incident wave characteristics. Since Stokes vector is the core measured compact polarimetric quantity capable of generating various indicators, such relation with backscattering components (i.e., target nature) offers an effective CP-based target classification scheme. It is evident that knowledge of the transmit wave is required for CP-based target classification. The model which is developed here is used to evaluate effects of the transmit wave variations on the CP classification end-products. In particular, variations of the incident wave ellipticity in the neighborhood of circular polarization (typical transmit polarization mode for CP applications) is modeled. It is also discussed that the circular polarization anomaly can be estimated using the ground truth for the purpose of CP classification product calibrations.

5. REFERENCES

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