Dual-Polarized Planar Phased Array Analysis for Meteorological Applications

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.


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  DR be on the order of 0.1 dB.In addition, it is desirable that the copolar correlation coefficient  ℎV error be less than 0.01.In previous research [9][10][11][12][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 crosspolarization, 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 ⃗  1 from the horizontal () port and ⃗  2 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.

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 as where {a ℎ , a V } is the horizontal and so-called "vertical" polarization basis and a  , a  , and a  are unit vectors in the spherical coordinate system.
We consider the array has a 90 ∘ angular range in azimuth and a 30 ∘ range in elevation, which is applicable for weather observations.Thus, in Figure 1  is from −45 ∘ to 45 ∘ and  is from 60 ∘ to 90 ∘ .For a well-designed array, it would be symmetrical with respect to .So in this paper we only consider  from 0 ∘ to 45 ∘ and  from 60 ∘ to 90 ∘ .Accordingly, the beam direction (  ,   ) = (90 ∘ , 0 ∘ ) is the broadside of the array.

Element Pattern.
The element pattern in a dual-polarized phased array can be written as

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.
For each element, we use a 2 × 2 complex matrix A to model the channel imbalance and channel isolation for the transmission while B is for the reception.A and B are written as where where (  ,   ) is the beam direction.  (  ,   ) and   (  ,   ) are weighting coefficients with respect to each element.A  and B  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, T(, ) and R(, ) reduce to where f ave (, ) is called the array average element pattern.
The subscripts  and  in ( 8) and ( 9) represent the transmission and reception, respectively.F  and F  are written as

Array Analysis in ATSR Mode
3.1.Formulation.For a point target with the polarization scattering matrix (PSM) S in the direction (, ) at the range , the received voltage matrix can be written as where  is a gain term.Here the superscript "" means matrix transpose. = 2/ and  is the wavelength.E inc ATSR is the unit excitation for  and  ports, which is written as The received voltage matrix for distributed precipitations can be expressed as an integral.Consider where Ω is the solid angle and Ω = sin  .In (13), the gain term  and the term related to range  are dropped for the sake of simplicity.To retrieve S(, ), the calibrated voltage matrix can be expressed as where the calibration matrices C  and C  are expressed as C  and C  can be obtained through array pattern measurements.By defining the calibrated voltage matrix is written as Assuming the intrinsic differential reflectivity  DR is defined as where ⟨⋅⟩ means the ensemble average.The bias of  DR can be calculated as where   (,  = ℎ, V) is the received power.Meanwhile, the integrated cross-polarization ratio (ICPR) is calculated as which is the minimal linear depolarization ratio  DR that can be measured by a weather radar.After some trivial mathematical derivations, we get where In ( 23)  ℎV is called the copolar correlation coefficient and  DP represents the differential phase.The symbol * means complex conjugate.
According to (22), it is clear that   DR and ICPR are related to both the array patterns and the intrinsic  dr .The impacts of  dr and  ℎV exp( DP ) on   DR 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  dr = 1 and  ℎV exp( DP ) = 1 so that we can focus on the biases due to the radar system.

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 R and T are written as where  uni (, ) and  tay (, ) are the array factors of the uniform and Taylor weightings.According to the definitions of C  and C  , T and R can be modeled as where   are the error terms after the calibration.Funi and Ftay 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 as where   ,   , and   are complex numbers.The unit of  and  is radian.If C  and C  have no error, there will be   = 0 at (  ,   ); that is,   = 0.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  DR = 0 dB and  DR = −∞ dB.So we can ignore the impacts of the finite beamwidth and sidelobes.Consequently,   DR can be calculated as DR 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 Δ and ICPR: ICPR ≈ 20 log (Δ) .
(37)  From Figures 4 and 5, we know that the calibration error   has great impacts on   DR and ICPR.For a single spherical scatterer, to achieve |  DR | < 0.1 dB, Δ should be less than 0.01, which is really demanding for antenna pattern measurements.Moreover, we see that the relation between Δ and |  DR | is linear while the relation between Δ and ICPR 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 2. Generate   ,   , and   through a random number generator.
The simulation parameters are listed in Table 2. (, ) means the uniform distribution in [, ] and Arg() represents the phase of .It should be pointed out that a −40 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 −54 dB which is equal to that of the WSR-88D.Thus the simulated results with a −13 dB uniform weighting and −40 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 as The calibrated pattern fdipole (, ) is then written as   DR | 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   DR is not obvious and   DR 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 ICPR is obvious, with about a 2.5 dB difference between the broadside and the beam direction (60 ∘ , 45 ∘ ).Furthermore, the ICPR of −32.6 dB at the broadside in Figure 7 is much larger than that of −40 dB calculated from (37), indicating that the finite beamwidth considerably affects the measurement of  DR .
We then set |  | = 0.02, |  |, |  | ∈ (0, 4) 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 |  DR | 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 (60 ∘ , 45 ∘ ) is about 2.5 dB and the minimal ICPR at the broadside (90 ∘ , 0 ∘ ) is about −26.5 dB, increasing by about 7.5 dB compared with that calculated from the approximation of (37).International Journal of Antennas and Propagation

Array with Imperfect H/V Channels.
With imperfect / channels, R and T can be written as A and B are expressed as First, we analyze the case with a single spherical scatterer in the beam direction (  ,   ).Thus we just need to consider T(, )| =  ,=  and R(, )| =  ,=  .Since   (  ,   ) and   (  ,   ) compensate the phase displacement exp( ⃗   ⋅ a  ), we can get As ∑  =1 ∑  =1 | X (  ,   )| = 1 and ∑  =1 ∑  =1 | Ŷ (  ,   )| = 1, the double summations on the right sides of (42) are close to the mathematical expectations of A and B. If  ℎV ,  Vℎ ,  ℎV ,  Vℎ ∼ (0, 2), we have  ℎV = 0,  Vℎ = 0,  ℎV = 0, and  Vℎ = 0.In this situation,   DR can be calculated as   In order to evaluate the bias under different conditions, we use the Monte Carlo simulation method again, which is shown below.
Step 2. Generate   ,   , and   from a random number generators and then calculate   .
Step 3. Generate A  and B  for each element.
Step 4. Calculate R and T.
Step 5. Calculate   DR and ICPR.).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 Δ is small.

Array Analysis in STSR Mode
In the STSR mode, the received voltages for distributed precipitations are expressed as where  ℎ () and  V () represent two waveforms from the  and  channels.Assuming  ℎV (, ) =  Vℎ (, ) = 0, (46) can be written as As shown in (47),  ℎ () is contaminated by both the firstand 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   DR < 0.1 dB.Hence, the accuracy requirement in the STSR mode is more demanding.
The orthogonal waveforms are usually employed to improve the polarimetric performance [20][21][22][23].The received voltages after passing the matched filters of the  and  channels can be written as where sℎ () =  ℎ  * ℎ ( 0 − ) and sV () =  V  * V ( 0 − ) are the matched filters of the  and  channels, respectively.⊗ means signal convolution.If  ℎ () and  V () are completely orthogonal, we can get In 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,  ℎ () and  V () cannot be completely orthogonal.Then we define Accordingly, the calibrated voltage matrix V() can be written as where C  and C  are defined in (15).Once the waveforms  ℎ () and  V () are known, P() can be calculated from (50).Then the calibration procedure in the STSR mode is still the same as that in the ATSR mode.

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  DR and  DR .  should be under 0.01 to achieve  DR < 0.1 dB, indicating that   should be under 1%.The imperfect  and  channels have considerable contributions to the biases of  DR and  DR .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   DR < 0.1 dB.The finite beamwidth considerably affects the measurement of  DR .However, the measurement of  DR 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  DR .Therefore, for a large array with a narrow beam that is commonly used for weather radars, the measurement performance of  DR can be directly estimated through the measurement at the boresight, which can significantly simplify the analysis for the measurement of  DR .
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.

Figure 1 :
Figure 1: The array configuration and spherical coordinate system used for radiated electric fields.

Figure 2 :
Figure 2: Dual-polarized / modules for polarimetric phased array weather radars.(a) is for the ATSR mode and (b) is for the STSR mode.In the ATSR mode, one channel is shared for transmitting  and  signals and two independent channels are used for reception.In the STSR mode, four independent channels, two for transmission, two for reception, and one / switch to commute the transmission and reception signals, are required.

Figure 13 :Figure 14 :
Figure 13: The impact of the channel coupling on |  DR |.

Table 3 :
The model parameters of a pair of crossed dipoles in the beam direction (60 ∘ , 45 ∘ ).

Table 3 ,
39)Choosing (  ,   ) = (60 ∘ , 45 ∘ ), we calculate   (, ).By using Matlab Curve Fitting Toolbox, we calculate the parameters   ,   , and   and show them in Table3. 2 is called the coefficient of determination, which is a number that indicates how well data fit a statistical model.As shown in  2 with respect to   is 0.99, indicating a very good approximation performance of the linear error model.According to Table3, we know   , ∼ (0, 2) 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 Table4.Actually, the practical phased array usually has an element spacing of about /2.Thus the ranges of

Table 4 :
The model parameters of a pair of crossed dipoles with the length of  in the beam direction (60 ∘ , 45 ∘ ). (deg)