Improved Calibration Technique for Quasi-Monostatic Polarimetric Measurement System Using a Dihedral as the Calibration Reference

The alluring polarimetric characteristics of a dihedral corner reflector make it specially useful as a polarimetric calibration reference. However, for a radar cross section (RCS) range with quasi-monostatic geometry where there exists a small bistatic angle, the polarimetric calibration error rapidly increases as the bistatic angle increases when a dihedral is used as the polarimetric calibration reference. In this work, an improved calibration technique for quasi-monostatic polarimetric measurement system is proposed. The scattering mechanism of a rectangular dihedral corner reflector in a quasi-monostatic radar system is analyzed, where the bistatic angle constrains to be within a few degrees. The polarimetric characteristic of the dihedral is evaluated by means of physical optics (POs) approximation. Combining the evaluation formulation with a nonlinear calibration technique, the polarimetric measurement error model can be adapted for the quasi-monostatic geometry so that accurate polarimetric calibration may be accomplished. The experimental results are presented to validate the proposed technique with essential improvement in the calibrated cross polarimetric measurements.


I. INTRODUCTION
HE problem of polarimetric calibration for monostatic measurement geometry has been widely studied. Since 1990s, the standard RST model [1] has been introduced to describe the errors of polarimetric measurement including the channel imbalances as well as antenna cross-talk. Wiesbeck [1], [2], Sarabandi, and Ulaby [3], [4] proposed different calibration techniques to obtain the accurate polarimetric scattering matrix (PSM) of targets for specific radar systems. Among others, much of the published works used a dihedral corner reflector as the calibration reference, such as the rectangular-shaped dihedral was used by Chen [5], Unal [6], and Muth [7], [8], the triangular-shaped dihedral was used by Gau [9] and Welsh [10]. In many radar cross section (RCS) ranges, the measurement geometry is quasi-monostatic where the transmitting antenna and receiving antenna are separated with a small distance, i.e., the measurement geometry is bistatic where the bistatic angle constraints to a few degrees. As is well known, the bistatic RCS approaches to the monostatic RCS when the bistatic angle is small [11]. However, the difference between monostatic [12], [13] and bistatic [14]- [17] RCSs of dihedral will result in additional calibration error. As a consequence, for accurate polarimetric calibration, quasi-monostatic correction must be made.
The objective of this study is to develop and recommend a modified polarimetric calibration procedure that can be used to correct the bistatic scattering error term of a dihedral for quasi-monostatic measurement geometries. This work is an essential extension of our previous conference paper on ICEAA 2016 [18], where the scattering mechanisms of a rectangular dihedral reflector with both monostatic and quasi-monostatic measurement geometries were analyzed and a correction formulation was proposed by taking the physical optics (PO) solution of a metal plate as reference. In the current work, we present a detailed formulation based on analytic physical optics (APO) [19], [20] for quasi-monostatic scattering of a rectangular dihedral rotating around radar line of sight (LOS). Using the derived formulation, the polarimetric measurement error model of dihedral with monostatic geometry is adapted to quasi-monostatic geometry. The system parameters are then obtained by means of Fourier analysis [7], [8], with more accurate polarimetric calibration being accomplished. Finally, experimental results are presented to validate the proposed calibration technique, demonstrating essential improvement of the cross polarimetric measurements.
The remainder of this paper is organized as follows. The error model of polarimetric measurement and calibration technique with monostatic geometry are briefly described in Section II. In Section III, the scattering mechanisms of dihedral reflector with quasi-monostatic geometry are analyzed and the evaluation formulation of the error caused by small bistatic angle is developed. The measurement error model of polarimetric calibration with quasi-monostatic geometry is discussed in Section IV, with a modified calibration technique being proposed for enhanced accuracy. Measurement and calibration results are presented in Section V with analysis to validate the proposed technique. We summarize the paper in Section VI.

II. POLARIMETRIC CALIBRATION TECHNIQUE FOR MONOSTATIC GEOMETRIES
Quite a few papers have been published that discuss the procedures performing full polarimetric calibration from radar PSM measurement. These procedures generally deal with systematic and linear errors. Statistical errors may be reduced by averaging over several measurement cycles. In the case considered, the orthogonal polarizations are linearly horizontal and vertical. The typical schematic block diagram used to describe the polarimetric measurement system is given by Fig.1  [1  After the vector background subtraction of isolation errors, the measured matrix can be normalized by the co-polarization channels and rewritten as are the cross-polarimetric coefficients of receiving and transmitting channels. The determination of these unknowns is based on the measurement of calibration references for which the PSMs are well known. In [1]- [10], different approaches for the calculation of the system error coefficients are shown.
Dihedral corner reflector is used as calibration reference in many calibration techniques. For a monostatic measurement geometry, the different orientations of a rectangular dihedral reflector [2] are shown in Fig.2 It is noted that in Fig.2, the y axis is along the radar LOS. The x and z axes are, respectively, the two cross-range coordinates.
The two plates forming the dihedral intersect at 90° and both of them are rectangles with w in width and h in height. The direction of the incident electromagnetic wave is perpendicular to the aperture of the dihedral. When the dihedral is rotated with an angle of θ about the LOS, as seen in Fig.2 , which depends on the dimensions of the dihedral and is assumed to be known, λ is the radar wavelength.
In most of the polarimetric calibration techniques, the dihedral need to be measured at two orientations such as with rotation angles of 0° and 45° [2], [5], [9], [10]. In other techniques, more polarimetric calibration data are measured during the dihedral rotation from 0° to 180° [6]- [8]. For the full polarimetric measurements, the measured values can be written as The matrix elements for all the polarization combinations have the general form where p and q are either h or v. We applied Fourier analysis to the data to determine the cross-polarimetric coefficients. Actually, there are some noises and errors that hardly be removed more or less in measurement. As a consequence, the measurement can then be expressed as the Fourier series Mathematically, the signals of dihedral correspond to the coefficients c 2 and s 2 , respectively. They are used to determine the cross-polarimetric coefficients, and others such as c 1 and s 1 are omitted in polarimetric calibration procedure. Because the background is stationary, the contributions of the background and noises to the dihedral signals are greatly reduced. The details of the theoretical expressions have been presented in [7] and [8]. With the measurement results of another calibration reference, all polarimetric error coefficients can be solved and the polarimetric calibration can be accomplished.

III. FORMULATION OF BISTATIC CORRECTION WITH QUASI-MONOSTATIC GEOMETRY
Many RCS ranges have quasi-monostatic measurement geometries where two antennas are used for transmitting and receiving, respectively, as seen in Fig.3. Here, suppose that the transmitting antenna is set on the right side while the receiving antenna is set on the left. It is seen that the incidence angle (half of the bistatic angle) is α and the scattering angle is -α, which are related to the locations of the antennas and the dihedral, while independent of the dihedral rotation and the polarization of the electromagnetic wave.
The small bistatic angle leads to the differences of scattering mechanisms between monostatic and quasi-monostatic geometries. Because of the special structure of dihedral reflector, the double reflection from the plates of the dihedral is the major scattering component during polarimetric calibration, which the electromagnetic wave reflect from one plate to another and then back in the scattering direction. Other components contributing to the scattered field are weaker than the double reflection component [12]. The differences of double reflections between monostatic and quasi-monostatic geometries are analyzed in detail. For monostatic measurement geometry, the incidence direction is normal sight, and the incidence angle equals to zero. The scattering direction of double reflection is parallel to the incidence direction so that it is right along the observing direction (LOS to the receiving antenna). For quasi-monostatic geometry, as shown in Fig.5, there is a misalignment between the scattering direction and the observing direction. The misalignment causes loss of energy received by antenna, which in other words, is bistatic magnitude correction term.
The scattering mechanism of a dihedral is similar to metal plate except that double reflections must be considered [21]. Therefore the metal plate with the same aperture size as the dihedral is analyzed to evaluate the bistatic correction term.
Specifically, the metal plate is w 2 in width and h in height.   6 shows the monostatic measurement geometry which irradiated in normal sight and incidence angle equals to zero. The scattering direction is similar to the dihedral with monostatic geometry. Fig.7 demonstrates the monostatic geometry with a small incidence angle. The incidence direction tilts a minus angle in horizontal, and the misalignment between scattering and observing directions occurs. It is similar to the dihedral with quasi-monostatic geometry. So we use it to evaluate the bistatic correction term of dihedral.
According to the PO solution, the RCS of metal plate is [21] where a and b are the width and height of the metal plate, respectively, φ is the incidence angle, k = 2π/λ is the wave number. Comparing with the normal sight situation, the correction term can be written as 2 2 ] sin The bistatic scattering of the dihedral corner reflector can be calculated by APO [19]. As the quasi-monostatic geometry shown in Fig.3, the APO solution can be approximated and simplified in small bistatic angle cases. A detailed derivation can be found in the Appendix. The correction term is given by (A10) in the Appendix and is duplicated here for convenience ] sin 2 It can be easily seen that the quasi-monostatic correction term of dihedral is the same as the correction term of plate, demonstrating the scattering mechanisms we analyzed. The bistatic RCS correction term of the dihedral reflector with rotation angle of 0° is

B. Dihedral with rotation angle of 90°
The scattering mechanism of dihedral with rotation angle of 90° is shown in Fig.8. The front view shows the double reflection of dihedral with quasi-monostatic geometry clearly. It can be easily seen that, in this situation, the scattering direction is the same as the observing direction. The RCS of dihedral is impacted by the reduction of effective illumination region. It can be omitted when the incidence angle approaches 0°. Finally, we think about the situation of rotation angle that is neither 0° nor 90°. Fig.9(a) shows the dihedral with rotation angle of θ with quasi-monostatic geometry.
Suppose that the dihedral is rotated from θ to 0° case. In order to keep the scattering characteristics unchanged, the locations of the antennas are rotated simultaneously with the bistatic angle fixed. It is shown in Fig.9(b) as the equivalent situation. The orientation of the dihedral is the same as that in Jackson's work [19], however, the polarization directions of electromagnetic waves are different. As a consequence, the scattering characteristics should be recalculated or revised on the basis of reference. According to (12), after small angle approximation, the bistatic correction term is not related to the polarization. Therefore, the bistatic angle can be decomposed to azimuth angle in horizontal and elevation angle in vertical. The bistatic correction term in horizontal is the same as 0° case and the azimuth angle is αcosθ. The bistatic correction term in vertical is the same as 90° case which can be ignored. In summary, when the rotation angle of the dihedral is θ, the bistatic correction term may be approximated by l PSM (αcosθ).

IV. MEASUREMENT ERROR MODEL WITH QUASI-MONOSTATIC GEOMETRY
In this section, by taking a specific example we discuss the measurement error model with quasi-monostatic geometry. We start from the characteristic of a dihedral rotating around the LOS, as seen in Fig. 10 and 11. The PSM of dihedral reflector is calculated using the method of moment (MOM) code of Feko software. The frequency is 10 GHz. Both plates of dihedral are 15 cm in width and 21 cm in height. The angle of incidence is 2° and the angle of observing is -2° for quasi-monostatic geometry. It is the case of the space between transmitting and receiving antennas is 50 cm and the target (dihedral) range is 7 m. The dihedral rotates from 0° to 175°, step by 5°. The real and imaginary parts of the vv polarization component are shown in Fig.10(a) and (b), respectively.
From the numerical result, it can be seen that the bistatic correction term varies with the rotation angle. It is the same as the analysis of scattering mechanisms and the calculation results in Section III. Specifically, when the rotation angle approaches to 0° or 180°, the effect of bistatic angle is strong and the difference between monostatic and quasi-monostatic is obvious. When the rotation angle approaches to 90°, there is little effect of bistatic angle and the difference between monostatic and quasi-monostatic is small.
The formulations in Section III can be applied here to eliminate the bistatic error of quasi-monostatic result, and the Fourier analysis is used to reduce the noise. On the other hand, the Fourier analysis can be applied to the quasi-monostatic result directly and the signal of dihedral can be acquired. It is shown as dot and dash line in Fig. 10. Because of the bistatic error of dihedral in most of the rotation positions, it can be easily seen that, the Fourier component is attenuated comparing with the monostatic result. The attenuation of Fourier analysis result should be estimated and corrected to ensure the accurate polarimetric calibration and it will be discussed in detail. As the evaluation formulations we get in Section III, the bistatic correction terms of dihedral reflector with rotation angles of 0°, 45°, 90° are l PSM (α), l PSM (αcos45°), 1, respectively. For the co-polarization, the attenuation of Fourier component is related to the bistatic errors of dihedral with rotation angles of 0° and 90°. Take vv polarization as an example, shown in the Appendix (A25), it will be written as The real and imaginary parts of the hv polarization component are shown in Fig.11(a) and (b), respectively.
For the cross-polarization, the response of the dihedral with rotation angle of 45° is almost stable during the Fourier analysis. It is used to represent the attenuation of Fourier component. Take hv polarization as an example, shown in the Appendix (A26), it will be written as The cosine term can be divided from the sine term to eliminate components related to the dihedral and amplification of antennas. A factor which only contains the crosspolarimetric coefficients and bistatic errors will be obtained. Similarly, for other polarimetric channels, the factors relate to cross-polarimetric coefficients are In the four equations above, there are four cross-polarimetric coefficients as unknown components. However, the four equations are not independent of each other, so the cross-polarimetric coefficients can not be determined. The measurement of other type polarimetric calibration reference is needed, such as the metal plate. The measurement value of a plate can be organized to We can choose three equations from the results of dihedral and combine with the equation of plate to obtain an equation group, such that the cross-polarimetric coefficients can be solved. According to |ε|<1 and other limits which are the real situations for most of the RCS system, the true solution can be picked out from the fault solutions.
Then, the gain of each polarimetric channel can be calculated. Take the result of hh polarization as an example, the gain A hh can be expressed as Till now, the four cross-factors and four channel amplifications of the RCS system are solved out. Finally the polarimetric calibration of target can be accomplished by

V. EXPERIMENTAL RESULTS
We use two rectangular dihedrals and two square metal plates with different size to perform polarimetric calibration in an indoor RCS measurement range. The big dihedral and plate are used as calibration references to calculate the system coefficients. The big dihedral consists of two plates sized 15 cm in width and 21 cm in height, and the square metal plate is 21.5 cm in width. The small dihedral and plate are used as targets whose PSMs are need to be calibrated. The small dihedral consists of two plates sized 7.5 cm in width and 10.5 cm in height, and the square metal plate is 11 cm in width. Scattering measurement is performed using a vector network analyzer (VNA). Time domain gating technique is applied to remove the possible target-room coupling during the measurement. The space between transmitting and receiving antennas is 19 cm and the target range is 4.5 m, that means the bistatic angle α is more than 1.2°. The measurement, Fourier result, and deviation of the reference dihedral are shown in Fig. 12(a) during the rotation of dihedral at 10 GHz. Altogether 400 data points were collected for each polarization during a full 360° rotation. In order to eliminate the effect of gains of the system, the results are normalized by the maximum of Fourier result and shown in Fig.  12(b). For an intensive study of the bistatic error, we enlarge the space between transmitting and receiving antennas to 40 cm, that means the bistatic angle α is more than 2.5°. The normalized results are presented in Fig. 13. From Fig. 12(b) and 13, it is found that larger differences do exist at the larger bistatic angle between measurement (quasi-monostatic) and Fourier results. The bistatic error will cause the attenuation of Fourier component compared with the monostatic result as discussed in Section IV. The measurement results also show that the larger bistatic angles change the shape of curve more obviously.
In Fig. 14(a) and (b), the results of target metal plate before and after polarimetric calibration are illustrated, respectively. It can be seen that the polarization isolation of the measurements is about 30 dB, while after polarimetric calibration, it is lowed to be about 45 dB.
In Fig. 15(a) and (b), the results of the target dihedral with 0° rotation before and after polarimetric calibration are presented, respectively. The dihedral with 0° rotation is a dominant co-polarization target and the cross-polarization component is about 50 dB down from the co-polarization component for the calibrated result, 20 dB better than the raw data. In this work, we proposed an improved polarimetric calibration technique for quasi-monostatic radar system, where the bistatic angle is constrained to be within a few degrees. The polarimetric characteristics of a rectangular dihedral corner reflector rotating around radar LOS in a quasi-monostatic radar system are studied. Based on the analysis of the scattering mechanisms for dihedral with both monostatic and quasimonostatic measurement geometries, the correction term caused by small bistatic angle is evaluated based on PO approximation. The polarimetric measurement error model is modified for the quasi-monostatic geometry so that accurate polarimetric calibration can be accomplished. By means of the derived formulation, the bistatic error of dihedral reflector can be greatly reduced. Experimental results show that the polarization isolation of calibrated data is well below 45 dB when the bistatic angle α is up to 1.2°. It is thus concluded that the proposed technique is useful for more accurate polarimetric calibration with quasi-monostatic measurement geometries, which are very common in RCS test ranges.

A. Analytic physical optics (APO) solution for dihedral with quasi-monostatic geometry
The APO solution for bistatic scattering from a dihedral has been calculated excellent by Jackson [19]. The APO result can be simplified according to the quasi-monostatic geometry. Specifically, the two faces of dihedral are same with w in width, the directions of both incidence and scattering are in x-y plane. Along the LOS, the transmitting antenna is set at the right hand while the receiving antenna is set at left. That means, for double reflection, the illumination area of dihedral is stable as only a near-boundary stripe at left can not be illuminated. For Considering that the incidence angle α constraints to a few degrees, the value of f(α) can be approximated to 1. The exponential components in (A1) and (A4) can be written as complex number ) sin sin cos (cos Similarly, the value of sinαsinA can be approximated to 0 as α is a small angle. The double reflection components of dihedral can be simplified as Comparing the quasi-monostatic result |S(α)| with the monostatic result |S(0)|, the correction term is can not be ignored and must be kept in the approximation of (A18), there will be cos6θ component in vv and sin6θ component in hv results. The simplified results of dihedral responses with quasi-monostatic geometry in this situation may be analyzed similarly.