#### Abstract

This paper presents a theoretical analysis for the accuracy requirements of the planar polarimetric phased array radar (PPPAR) in meteorological applications. Among many factors that contribute to the polarimetric biases, four factors are considered and analyzed in this study, namely, the polarization distortion due to the intrinsic limitation of a dual-polarized antenna element, the antenna pattern measurement error, the entire array patterns, and the imperfect horizontal and vertical channels. Two operation modes, the alternately transmitting and simultaneously receiving (ATSR) mode and the simultaneously transmitting and simultaneously receiving (STSR) mode, are discussed. For each mode, the polarimetric biases are formulated. As the STSR mode with orthogonal waveforms is similar to the ATSR mode, the analysis is mainly focused on the ATSR mode and the impacts of the bias sources on the measurement of polarimetric variables are investigated through Monte Carlo simulations. Some insights of the accuracy requirements are obtained and summarized.

#### 1. Introduction

Recently, the weather radar community has paid much attention to the polarimetric phased array radar (PPAR) due to its agile electronic beam steering capability, which has the potential to significantly advance weather observations [1]. Various system designs have been presented and studied. A low cost mobile X-band phased array weather radar with phase-tilt antenna array was developed in [2]. Fulton and Chappell [3] designed an S-band, differentially probe-fed, stacked patch antenna for multifunctional phased array weather radar applications and studied the calibration method [4]. Zhang et al. [5] proposed a cylindrical configuration for the polarimetric phased array weather radar and illustrated the advantages of the cylindrical configuration over the planar configuration. In [6] an overview of the calibration techniques, tools, and challenges surrounding the development of a cylindrical polarimetric phased array radar (CPPAR) demonstrator was provided. The design of interleaved sparse arrays [7] for the agile polarization control was analyzed with the purpose of meteorological applications. Dong et al. [8] analyzed the polarization characteristics of two ideal orthogonal Huygens sources and evaluated their polarimetry performance.

As shown in [1, 9], a high-accuracy measurement of polarimetric variables is required to provide meaningful information for reliable hydrometeor classifications and improved quantitative precipitation estimations. For example, it is desirable that the measurement error for the differential reflectivity be on the order of 0.1 dB. In addition, it is desirable that the copolar correlation coefficient error be less than 0.01. In previous research [9–13], the polarimetric biases of weather radars with mechanically scanning antennas have been widely discussed. A detailed literature review of the bias analysis and calibration methods was presented in [9]. Generally, in order to make accurate polarimetric measurements by using a mechanically scanning antenna, a narrow beam with low sidelobes, low coaxial cross-polarization, and high polarization isolation are indispensable.

Although the weather radar polarimetry has matured for years, there are some challenges for the planar polarimetric phased array radar (PPPAR) [14]. As shown in Figure 1, the array is placed on the plane. When the beam is away from the principle planes, the electric field from the horizontal () port and from the vertical port are not necessarily orthogonal, which will introduce polarimetric biases that are not negligible. The nonorthogonality of the and polarizations is called the polarization distortion in this paper. Meanwhile, the polarization distortion also includes mismatches in the power levels of and beams as a function of the scan angle. The calibration matrix that relies on the measured array patterns is needed to calibrate the polarimetric bias due to the polarimetric distortion. As the measured antenna pattern always contains measurement errors, the calibration matrix cannot completely calibrate the bias due to the polarization distortion, which is not thoroughly analyzed in previous research [1, 15, 16] and will be discussed in this study. Moreover, in [1, 15, 16] it implies that the beam is thin enough so that the calibration performed at the boresight is sufficient to retrieve the polarimetric variables. However, in practice the finite beamwidth also contributes to the polarimetric bias, which will be evaluated in this paper. Besides the antenna, the imperfect channels can still bias the polarimetric variables, which will be modeled and analyzed. Actually, other factors, such as the mismatch between element patterns, spatial variations of cross-polarization patterns, mutual coupling edge effects, diffracted fields, and surface waves, can significantly affect the overall accuracy of a PPPAR. To simplify the analysis, these factors are ignored.

Usually, there are two operation modes chosen for weather observations, the alternately transmitting and simultaneously receiving (ATSR) mode and the simultaneously transmitting and simultaneously receiving (STSR) mode. Each mode has its advantages and disadvantages. With a “perfect” antenna, the STSR mode is vastly superior to the ATSR mode in the worst-case polarimetric/spectral situations. Thus, the STSR mode is the preferred mode from a meteorological standpoint. However, both the theoretical analysis and measurement experiments have shown that the STSR mode has higher accuracy requirements than the ATSR mode. This paper is mainly focused on the ATSR mode as it is simple for the analysis.

The remainder of this paper is organized as follows. Section 2 presents the array model. Sections 3 and 4 give the detailed analysis in the ATSR and STSR modes, respectively. Summaries and conclusions are made in Section 5.

#### 2. Array Model

The coordinate system and array configuration are shown in Figure 1. It is common that in antenna measurements the antenna is placed on the plane. In this situation the and vectors correspond to the second definition in [17]. In this paper, the array with rows and columns is placed on the plane, which is different from the typical situation. The reason is that in meteorological applications when the array is placed on the plane, the expressions of the horizontal and vertical polarization basis are simple, which are written aswhere is the horizontal and so-called “vertical” polarization basis and , , and are unit vectors in the spherical coordinate system.

We consider the array has a angular range in azimuth and a range in elevation, which is applicable for weather observations. Thus, in Figure 1 is from to and is from to . For a well-designed array, it would be symmetrical with respect to . So in this paper we only consider from to and from to . Accordingly, the beam direction is the broadside of the array.

##### 2.1. Element Pattern

The element pattern in a dual-polarized phased array can be written aswhere (i) is the horizontal electric field component when only the port is excited(ii) is the horizontal electric field component when only the port is excited(iii) is the vertical electric field component when only the port is excited(iv) is the vertical electric field component when only the port is excited.For a practical dual-polarized antenna element, the cross-polarization components and are not 0 in the beam scan area.

##### 2.2. Transmission and Reception Patterns

Figure 2 from [18] shows dual-polarized modules for polarimetric phased array weather radars in the ATSR and STSR modes. As explained in [4], the module connected to each element may have cross-coupling between its and channels as well as complex gain/phase imbalances. These cross-couplings and imbalances can be modeled by a matrix multiplication of the and signals presented to the module on both transmission and reception with components as designated in Figure 3. In this paper, we use the term “channel isolation” to express the cross-coupling between the and channels. Meanwhile, we use the term “channel imbalance” to express complex gain/phase imbalances between the and channels.

**(a)**

**(b)**

For each element, we use a complex matrix to model the channel imbalance and channel isolation for the transmission while is for the reception. and are written aswhere , , , and describe the channel imbalance and , , , and give the channel isolation. For simplicity, we assume . Then the channel imbalance CIM is defined as Note that if , there will be . Thus we use the expression such that CIM remains positive.

Similarly, the channel isolation CIS can be defined aswhich means CIS is defined as the worst value among , , , and .

The array transmission pattern and reception pattern are expressed as [19] where is the beam direction. and are weighting coefficients with respect to each element. and model the imperfect channel effects.

The mutual coupling between array elements is complicated so that a thorough analysis of the mutual coupling usually includes the full-wave electromagnetic computation and measurement experiments, which is beyond the scope of this paper. Moreover, for a large array, most of the elements are far from an edge. Therefore, except for the phase center displacement, all of the central element patterns are nearly the same. So it is reasonable to use the array average element pattern to replace the single element pattern. Hence, and reduce towhere is called the array average element pattern. The subscripts and in (8) and (9) represent the transmission and reception, respectively. and are written as

#### 3. Array Analysis in ATSR Mode

##### 3.1. Formulation

For a point target with the polarization scattering matrix (PSM) in the direction at the range , the received voltage matrix can be written aswhere is a gain term. Here the superscript “” means matrix transpose. and is the wavelength. is the unit excitation for and ports, which is written as

The received voltage matrix for distributed precipitations can be expressed as an integral. Considerwhere is the solid angle and . In (13), the gain term and the term related to range are dropped for the sake of simplicity. To retrieve , the calibrated voltage matrix can be expressed aswhere the calibration matrices and are expressed as and can be obtained through array pattern measurements. By definingthe calibrated voltage matrix is written as

Assuming

the intrinsic differential reflectivity is defined aswhere means the ensemble average. The bias of can be calculated aswhere () is the received power. Meanwhile, the integrated cross-polarization ratio (ICPR) is calculated aswhich is the minimal linear depolarization ratio that can be measured by a weather radar. After some trivial mathematical derivations, we getwhereIn (23) is called the copolar correlation coefficient and represents the differential phase. The symbol means complex conjugate.

According to (22), it is clear that and ICPR are related to both the array patterns and the intrinsic . The impacts of and on have been thoroughly analyzed in [9] and those conclusions can be directly applied for the analysis of a PPPAR. Hence, in this paper we assume and so that we can focus on the biases due to the radar system.

##### 3.2. Array with Perfect H/V Channels

An array with perfect channels means that the channel imbalance and isolation can be ignored. Hence, the transmission and reception patterns can be written as

For the transmission pattern, the radiation power is principal. Thus, a uniform illumination is applied. For the reception pattern, a beam with low sidelobes is desired. Here, we choose the Taylor weighting. So and are written aswhere and are the array factors of the uniform and Taylor weightings. According to the definitions of and , and can be modeled aswhere are the error terms after the calibration. and are the normalized array factors. The superscripts and represent the transmission and reception, which are usually dropped for simplicity. To simplify the analysis, is modeled aswhere , , and are complex numbers. The unit of and is radian. If and have no error, there will be at ; that is, . However, due to the antenna pattern measurement errors, is not 0. In addition, and indicate the polarization variation near . It should be pointed out that the linear error model (34) is most appropriate for well-behaved elements making up an array that is large enough to ensure that it is accurate over the beamwidth of the overall array.

According to Appendix, we know that the upper bound of has the same level as the relative error upper bound of the antenna measurements. Therefore, in the rest of this paper we just focus on .

First, we analyze a simple case to get some insights towards . We assume there is only one spherical scatterer in the beam direction , indicating dB and dB. So we can ignore the impacts of the finite beamwidth and sidelobes. Consequently, can be calculated asFurthermore, we assume and the phase of is uniformly distributed in . Using Taylor expansion and ignoring the second and higher order terms, we can get the approximation of the average bias of :where means mathematical expectation. Since has a symmetric distribution centered at 0, there is . Hence, we use other than . Figure 4 shows the relation between and . The red line is calculated from (36) while the blue line is obtained through Monte Carlo simulation in which and are generated from a random number generator and is calculated from (35). In Figure 4 we see that the approximation from (36) agrees well with the result from Monte Carlo simulation.

Using the same procedure, the average ICPR is derived in (37). Figure 5 shows the relation between and :

From Figures 4 and 5, we know that the calibration error has great impacts on and ICPR. For a single spherical scatterer, to achieve dB, should be less than , which is really demanding for antenna pattern measurements. Moreover, we see that the relation between and is linear while the relation between and is logarithmic.

As revealed in [9, 12, 13], the finite beamwidth has considerable impacts on the measurement of polarimetric variables. In order to evaluate the bias under different conditions, a method based on Monte Carlo simulation is developed so that we can evaluate the polarimetric bias with different parameters. The array parameters are shown in Table 1. The simulation procedure is shown below.

*Step 1. *Specify the polarization distortion calibration error .

*Step 2. *Generate , , and through a random number generator.

*Step 3. *Calculate from (32) and from (33).

*Step 4. *Calculate from (17).

*Step 5. *Calculate from (20) and ICPR from (21).

The simulation parameters are listed in Table 2. means the uniform distribution in and represents the phase of . It should be pointed out that a dB Taylor weighting is not practical for the implementation. According to [14], for weather observations the two-way sidelobe level of a PPPAR is expected to be under dB which is equal to that of the WSR-88D. Thus the simulated results with a dB uniform weighting and dB Taylor weighting are more comparable to those of radars with mechanical scanning antennas.

The ranges of and in Table 2 are determined based on the radiation pattern of a pair of crossed dipoles, which is written asThe calibrated pattern is then written asChoosing , we calculate . By using Matlab Curve Fitting Toolbox, we calculate the parameters , , and and show them in Table 3. is called the coefficient of determination, which is a number that indicates how well data fit a statistical model. As shown in Table 3, with respect to is 0.99, indicating a very good approximation performance of the linear error model. According to Table 3, we know is valid. Using the same procedure, we calculate the parameters , , and for a pair of crossed dipoles with the length of and show them in Table 4. Actually, the practical phased array usually has an element spacing of about . Thus the ranges of and should be better than the worst value in Table 4. In this paper, we assume and .

Figures 6 and 7 show the simulated and in the beam scan area . In Figure 6 most of are between 0.095 dB and 0.105 dB, which agrees with the approximation of (36). Figure 6 indicates that the impact of the finite beamwidth on is not obvious and is not sensitive to the beam expansion due to the beam scan. On the contrary, in Figure 7 the impact of the beam expansion on is obvious, with about a 2.5 dB difference between the broadside and the beam direction . Furthermore, the ICPR of dB at the broadside in Figure 7 is much larger than that of dB calculated from (37), indicating that the finite beamwidth considerably affects the measurement of .

We then set , and keep other parameters the same as those in Tables 1 and 2. Figures 8 and 9 show the simulation results. In Figure 8, most of are between 0.19 dB and 0.21 dB, which also agrees with (36) very well. In Figure 9, the difference between the broadside and the beam direction is about 2.5 dB and the minimal ICPR at the broadside is about dB, increasing by about 7.5 dB compared with that calculated from the approximation of (37).

##### 3.3. Array with Imperfect H/V Channels

With imperfect channels, and can be written as and are expressed as

First, we analyze the case with a single spherical scatterer in the beam direction . Thus we just need to consider and . Since and compensate the phase displacement , we can getAs and , the double summations on the right sides of (42) are close to the mathematical expectations of and . If , we have , , , and . In this situation, can be calculated asIf , (43) reduces toAssuming , based on the definition of CIM in (5), (44) can be expressed as

Figure 10 shows the simulated of a single spherical scatterer based on (43), in which , , , , and . In Figure 10 when , almost remains constant around 0.26 dB and when , increases linearly with a slope less than 10. When is small the channel imbalance has the main contribution to . Otherwise, the polarization distortion calibration error is the dominant bias source.

In order to evaluate the bias under different conditions, we use the Monte Carlo simulation method again, which is shown below.

*Step 1. *Specify .

*Step 2. *Generate , , and from a random number generators and then calculate .

*Step 3. *Generate and for each element.

*Step 4. *Calculate and .

*Step 5. *Calculate and ICPR.

Figures 11 and 12 show simulated and in the beam direction , in which , , , and . In Figure 11, the results with the finite beamwidth match the results of a single spherical scatterer well. However, Figure 12 indicates the finite beamwidth considerably affects ICPR when is small.

We then set , and keep other parameters the same as those in Figures 11 and 12 to evaluate the impact of the channel isolation. The simulated and considering the imperfect channel isolation are given in Figures 13 and 14 with the blue squares while the simulated and from Figures 11 and 12 are still shown with the red triangles. In Figure 13 the impact of the imperfect channel isolation on is not obvious. However, in Figure 14, the impact on ICPR is obvious when is small.

#### 4. Array Analysis in STSR Mode

In the STSR mode, the received voltages for distributed precipitations are expressed aswhere and represent two waveforms from the and channels. Assuming , (46) can be written asAs shown in (47), is contaminated by both the first- and second-order terms of the cross-polar patterns. In the ATSR mode, the received voltages are just contaminated by the second-order terms of the cross-polar patterns. Thus the accuracy requirement in the STSR mode should be higher than that in the ATSR mode. As discussed in Section 3, in the ATSR mode, the relative error of the antenna pattern measurement should be under 1% to achieve dB. Hence, the accuracy requirement in the STSR mode is more demanding.

The orthogonal waveforms are usually employed to improve the polarimetric performance [20–23]. The received voltages after passing the matched filters of the and channels can be written aswhere and are the matched filters of the and channels, respectively. means signal convolution. If and are completely orthogonal, we can getIn this situation, (48) is equivalent to (13) derived in the ATSR mode. Thus the same calibration procedure and analysis in Section 3 can be applied.

In practice, and cannot be completely orthogonal. Then we defineSo (48) can be written asAccordingly, the calibrated voltage matrix can be written aswhere and are defined in (15).

Once the waveforms and are known, can be calculated from (50). Then the calibration procedure in the STSR mode is still the same as that in the ATSR mode.

#### 5. Conclusions

In this paper, we analyze the accuracy requirements of a PPPAR in the ATSR and STSR modes. Among many factors, we focus on the polarization distortion due to the intrinsic limitation of a dual-polarized antenna element, the antenna pattern measurement error, the entire array patterns, and the imperfect channels. Other factors such as the mutual coupling between the array elements are also important for the accurate weather measurement. However, these factors are ignored to simplify the analysis in this study.

The polarization distortion calibration error that has the same level as the relative error upper bound of the antenna pattern measurement is critical for the biases of and . should be under 0.01 to achieve dB, indicating that should be under 1%. The imperfect and channels have considerable contributions to the biases of and . The channel isolation CIS should be over 40 dB so that the impact of the channel isolation is negligiable. According to (45), the channel imbalance CIM should be under 0.05 dB to ensure dB.

The finite beamwidth considerably affects the measurement of . However, the measurement of is not sensitive to the finite beamwidth. Moreover, the impact of the beam scan that results in the beam expansion is just obvious on the measurement of . Therefore, for a large array with a narrow beam that is commonly used for weather radars, the measurement performance of can be directly estimated through the measurement at the boresight, which can significantly simplify the analysis for the measurement of .

In the STSR mode, if orthogonal waveforms are applied, the analysis is the same as that in the ATSR mode. Otherwise, the measurement performance may be worse than that in the ATSR mode. In the future research, those ignored factors could be taken into account to have a better understanding about the accuracy requirements of a PPPAR.

#### Appendix

#### Polarization Distortion Calibration Error

The measured element pattern can be expressed aswhere represents the absolute measurement error and is the real element pattern. It should be pointed out that the measured element pattern does not include the errors associated with the imperfect and channels which are characterized by the matrices and in (3) and (4), respectively. The matrices and are integrated in the array transmission and reception patterns to account for the bias contributions of the imperfect and channels. Therefore, we can just focus on the measurement error of .

The calibrated element pattern can be expressed asBy using the Matrix Inversion Lemma, we can getSubstituting (A.3) in (A.2), we can getwhere is the identity matrix. Thus the polarization distortion calibration error can be written asEquation (A.5) shows that the polarization distortion calibration error is related to both the absolute measurement error and the element pattern itself.

For a well-designed antenna element, the copolar patterns are larger than the cross-polar patterns. In addition, since , the matrix elements of are far less than the matrix elements of . Thus, (A.5) can be approximated asIt should be pointed out that (A.6) is only valid under the condition that the matrix elements of are far less than the matrix elements of . Fortunately, this condition is usually satisfied for a well-designed antenna element in the beam scan area.

Since the relative error is more essential to represent the measurement accuracy than the absolute error, we define the relative error upper bound of asAccording to (A.6), the polarization distortion calibration error can be estimated asConsequently, we can get

For a well-designed antenna element, (A.9) reveals that and are at the same level. Thus we just focus on the polarization distortion calibration error in this paper. When we need to focus on a specific array, the analysis herein can be applied.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant nos. 61201330, 61490690, and 61490694. The authors sincerely express their gratitude to the anonymous reviewers.