# Dual-Polarized Planar Phased Array Analysis for Meteorological Applications

Authors: Chen Pang; Peter Hoogeboom; François Le Chevalier; Herman W. J. Russchenberg; Jian Dong; Tao Wang; Xuesong Wang
Journal: International Journal of Antennas and Propagation
 (2015)

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## 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.


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## Body

## 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 ZDR be on the order of 0.1 dB. In addition, it is desirable that the copolar correlation coefficient ρhv 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 yz plane. When the beam is away from the principle planes, the electric field E→1 from the horizontal (H) port and E→2 from the vertical (V) port are not necessarily orthogonal, which will introduce polarimetric biases that are not negligible. The nonorthogonality of the H and V polarizations is called the polarization distortion in this paper. Meanwhile, the polarization distortion also includes mismatches in the power levels of H and V 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 H/V 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.Figure 1
The array configuration and spherical coordinate system used for radiated electric fields.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. Section2 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 Figure1. It is common that in antenna measurements the antenna is placed on the xy plane. In this situation the H and V vectors correspond to the second definition in [17]. In this paper, the array with M rows and N columns is placed on the yz plane, which is different from the typical situation. The reason is that in meteorological applications when the array is placed on the yz plane, the expressions of the horizontal and vertical polarization basis are simple, which are written as(1)ah=aφ,av=-aθ,where {ah,av} is the horizontal and so-called “vertical” polarization basis and ar, aθ, and aφ are unit vectors in the spherical coordinate system.We consider the array has a90∘ angular range in azimuth and a 30∘ range in elevation, which is applicable for weather observations. Thus, in Figure 1  φ is from -45∘ to 45∘ and θ is from 60∘ to 90∘. For a well-designed array, it would be symmetrical with respect to φ. So in this paper we only consider φ from 0∘ to 45∘ and θ from 60∘ to 90∘. Accordingly, the beam direction (θS,φS)=(90∘,0∘) is the broadside of the array.


### 2.1. Element Pattern

The element pattern in a dual-polarized phased array can be written as(2)fθ,φ=fhhθ,φfhvθ,φfvhθ,φfvvθ,φ,where (i)
fhh(θ,φ) is the horizontal electric field component when only the H port is excited(ii)
fhv(θ,φ) is the horizontal electric field component when only the V port is excited(iii)
fvh(θ,φ) is the vertical electric field component when only the H port is excited(iv)
fvv(θ,φ) is the vertical electric field component when only the V port is excited.For a practical dual-polarized antenna element, the cross-polarization components fhv(θ,φ) and fvh(θ,φ) are not 0 in the beam scan area.


### 2.2. Transmission and Reception Patterns

Figure2 from [18] shows dual-polarized T/R modules for polarimetric phased array weather radars in the ATSR and STSR modes. As explained in [4], the T/R module connected to each element may have cross-coupling between its H and V channels as well as complex gain/phase imbalances. These cross-couplings and imbalances can be modeled by a matrix multiplication of the H and V signals presented to the T/R 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 H and V channels. Meanwhile, we use the term “channel imbalance” to express complex gain/phase imbalances between the H and V channels.Figure 2
Dual-polarizedT/R 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 H and V signals and two independent channels are used for reception. In the STSR mode, four independent channels, two for transmission, two for reception, and one T/R switch to commute the transmission and reception signals, are required.
(a)
(b)Figure 3
Channel imbalance and channel coupling.For each element, we use a2×2 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(3)A=ahhahvavhavv,(4)B=bhhbhvbvhbvv,where ahh, avv, bhh, and bvv describe the channel imbalance and ahv, avh, bhv, and bvh give the channel isolation. For simplicity, we assume ahh=bhh=1. Then the channel imbalance CIM is defined as (5)CIM=max⁡20log⁡1avv,20log⁡1bvv.Note that if avv>1, there will be 20log⁡(1/avv)<0. Thus we use the expression 20log⁡(1/avv)⁡ such that CIM remains positive.Similarly, the channel isolation CIS can be defined as(6)CIS=20log⁡max⁡1avh,avvahv,1bvh,bvvbhvwhich means CIS is defined as the worst value among 1/|avh|, |avv|/|ahv|, 1/|bvh|, and |bvv|/|bhv|.The array transmission patternT(θ,φ) and reception pattern R(θ,φ) are expressed as [19] (7)Tθ,φ=∑m=1M∑n=1NXmnθS,φSfmnθ,φ·Amn,Rθ,φ=∑m=1M∑n=1NYmnθS,φSfmnθ,φ·Bmn,where (θS,φS) is the beam direction. Xmn(θS,φS) and Ymn(θS,φS) are weighting coefficients with respect to each element. Amn and Bmn 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(θ,φ) and R(θ,φ) reduce to(8)Tθ,φ=faveθ,φFT,(9)Rθ,φ=faveθ,φFR,where fave(θ,φ) is called the array average element pattern. The subscripts T and R in (8) and (9) represent the transmission and reception, respectively. FT and FR are written as (10)FT=∑m=1M∑n=1Nexp⁡jkr→mn·arXmnθS,φSAmn,FR=∑m=1M∑n=1Nexp⁡jkr→mn·arYmnθS,φSBmn.


## 2.1. Element Pattern

The element pattern in a dual-polarized phased array can be written as(2)fθ,φ=fhhθ,φfhvθ,φfvhθ,φfvvθ,φ,where (i)
fhh(θ,φ) is the horizontal electric field component when only the H port is excited(ii)
fhv(θ,φ) is the horizontal electric field component when only the V port is excited(iii)
fvh(θ,φ) is the vertical electric field component when only the H port is excited(iv)
fvv(θ,φ) is the vertical electric field component when only the V port is excited.For a practical dual-polarized antenna element, the cross-polarization components fhv(θ,φ) and fvh(θ,φ) are not 0 in the beam scan area.


## 2.2. Transmission and Reception Patterns

Figure2 from [18] shows dual-polarized T/R modules for polarimetric phased array weather radars in the ATSR and STSR modes. As explained in [4], the T/R module connected to each element may have cross-coupling between its H and V channels as well as complex gain/phase imbalances. These cross-couplings and imbalances can be modeled by a matrix multiplication of the H and V signals presented to the T/R 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 H and V channels. Meanwhile, we use the term “channel imbalance” to express complex gain/phase imbalances between the H and V channels.Figure 2
Dual-polarizedT/R 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 H and V signals and two independent channels are used for reception. In the STSR mode, four independent channels, two for transmission, two for reception, and one T/R switch to commute the transmission and reception signals, are required.
(a)
(b)Figure 3
Channel imbalance and channel coupling.For each element, we use a2×2 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(3)A=ahhahvavhavv,(4)B=bhhbhvbvhbvv,where ahh, avv, bhh, and bvv describe the channel imbalance and ahv, avh, bhv, and bvh give the channel isolation. For simplicity, we assume ahh=bhh=1. Then the channel imbalance CIM is defined as (5)CIM=max⁡20log⁡1avv,20log⁡1bvv.Note that if avv>1, there will be 20log⁡(1/avv)<0. Thus we use the expression 20log⁡(1/avv)⁡ such that CIM remains positive.Similarly, the channel isolation CIS can be defined as(6)CIS=20log⁡max⁡1avh,avvahv,1bvh,bvvbhvwhich means CIS is defined as the worst value among 1/|avh|, |avv|/|ahv|, 1/|bvh|, and |bvv|/|bhv|.The array transmission patternT(θ,φ) and reception pattern R(θ,φ) are expressed as [19] (7)Tθ,φ=∑m=1M∑n=1NXmnθS,φSfmnθ,φ·Amn,Rθ,φ=∑m=1M∑n=1NYmnθS,φSfmnθ,φ·Bmn,where (θS,φS) is the beam direction. Xmn(θS,φS) and Ymn(θS,φS) are weighting coefficients with respect to each element. Amn and Bmn 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(θ,φ) and R(θ,φ) reduce to(8)Tθ,φ=faveθ,φFT,(9)Rθ,φ=faveθ,φFR,where fave(θ,φ) is called the array average element pattern. The subscripts T and R in (8) and (9) represent the transmission and reception, respectively. FT and FR are written as (10)FT=∑m=1M∑n=1Nexp⁡jkr→mn·arXmnθS,φSAmn,FR=∑m=1M∑n=1Nexp⁡jkr→mn·arYmnθS,φSBmn.


## 3. Array Analysis in ATSR Mode

### 3.1. Formulation

For a point target with the polarization scattering matrix (PSM)S in the direction (θ,φ) at the range r, the received voltage matrix can be written as(11)V=Cexp⁡-j2kr4πr2Rt·S·T·EATSRinc,where C is a gain term. Here the superscript “t” means matrix transpose. k=2π/λ and λ is the wavelength. EATSRinc is the unit excitation for H and V ports, which is written as(12)EATSRinc=1001.The received voltage matrix for distributed precipitations can be expressed as an integral. Consider(13)V=∫ΩRtθ,φ·Sθ,φ·Tθ,φdΩ,where Ω is the solid angle and dΩ=sin⁡θdθdφ. In (13), the gain term C and the term related to range r are dropped for the sake of simplicity. To retrieve S(θ,φ), the calibrated voltage matrix can be expressed as(14)V^=CR·V·CT=∫ΩCR·Rt·S·T·CTdΩ,where the calibration matrices CT and CR are expressed as(15)CT=Tθ,φθ=θS,φ=φS-1,CR=Rtθ,φθ=θS,φ=φS-1.CT and CR can be obtained through array pattern measurements. By defining(16)R^=R·CRt=R^hhR^hvR^vhR^vv,T^=T·CT=T^hhT^hvT^vhT^vv,the calibrated voltage matrix is written as(17)V^=∫ΩR^t·S·T^dΩ.Assuming(18)Sθ,φ=Shhθ,φ00Svvθ,φthe intrinsic differential reflectivityZDR is defined as(19)ZDR=10log⁡Shh2Svv2,where · means the ensemble average. The bias of ZDR can be calculated as(20)ZDRb=10log⁡PhhPvv-ZDR,where Pij (i,j=h,v) is the received power. Meanwhile, the integrated cross-polarization ratio (ICPR) is calculated as(21)ICPR=10log⁡PvhPhhwhich is the minimal linear depolarization ratio LDR that can be measured by a weather radar. After some trivial mathematical derivations, we get(22)ZDRb=10log⁡1+W1Zdr-1/2W2+W3Zdr1/2,ICPR=10log⁡W4+W5Zdr-1+W6Zdr-1/21+W1Zdr-1/2,where(23)ρhvexp⁡jϕDP=Shh∗SvvShh2Svv2,(24)W1=∫Ω2ReT^vhR^vhT^hh∗R^hh∗ρhvexp⁡jϕDPdΩ∫ΩThh2Rhh2dΩ,(25)W2=∫ΩT^vv2R^vv2dΩ∫ΩThh2Rhh2dΩ,(26)W3=∫Ω2ReT^vvR^vvT^hv∗R^hv∗ρhvexp⁡jϕDPdΩ∫ΩThh2Rhh2dΩ,(27)W4=∫ΩThh2Rhv2dΩ∫ΩThh2Rhh2dΩ,(28)W5=∫ΩTvh2Rvv2dΩ∫ΩThh2Rhh2dΩ,(29)W6=∫Ω2ReTvhRvvThh∗Rhv∗ρhvexp⁡jϕDPdΩ∫ΩThh2Rhh2dΩ.In (23) ρhv is called the copolar correlation coefficient and ϕDP represents the differential phase. The symbol ∗ means complex conjugate.According to (22), it is clear that ZDRb and ICPR are related to both the array patterns and the intrinsic Zdr. The impacts of Zdr and ρhvexp⁡(jϕDP) on ZDRb 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 Zdr=1 and ρhvexp⁡(jϕDP)=1 so that we can focus on the biases due to the radar system.


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

An array with perfectH/V channels means that the channel imbalance and isolation can be ignored. Hence, the transmission and reception patterns can be written as(30)Tθ,φ=faveθ,φ∑m=1M∑n=1Nexp⁡jkr→mn·arXmnθS,φS,Rθ,φ=faveθ,φ∑m=1M∑n=1Nexp⁡jkr→mn·arYmnθS,φS.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. SoR^ and T^ are written as(31)T^=T·CT=Funiθ,φfaveθ,φ·CT,R^=R·CRt=Ftayθ,φfaveθ,φ·CRt,where Funi(θ,φ) and Ftay(θ,φ) are the array factors of the uniform and Taylor weightings. According to the definitions of CT and CR, T^ and R^ can be modeled as(32)T^=F^uni1+εhhTθ,φεhvTθ,φεvhTθ,φ1+εvvTθ,φ,(33)R^=F^tay1+εhhRθ,φεhvRθ,φεvhRθ,φ1+εvvRθ,φ,where εij are the error terms after the calibration. F^uni and F^tay are the normalized array factors. The superscripts T and R represent the transmission and reception, which are usually dropped for simplicity. To simplify the analysis, εij is modeled as(34)εijθ,φ≈αijθ-θS+βijφ-φS+δij,where αij, βij, and δij are complex numbers. The unit of θ and φ is radian. If CT and CR have no error, there will be εij=0 at (θS,φS); that is, δij=0. However, due to the antenna pattern measurement errors, δij is not 0. In addition, αij and βij indicate the polarization variation near (θS,φS). 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 ofδij has the same level as the relative error upper bound Ef of the antenna measurements. Therefore, in the rest of this paper we just focus on δij.First, we analyze a simple case to get some insights towardsδij. We assume there is only one spherical scatterer in the beam direction (θS,φS), indicating ZDR=0dB and LDR=-∞ dB. So we can ignore the impacts of the finite beamwidth and sidelobes. Consequently, ZDRb can be calculated as(35)ZDRb=20log⁡1+δhhT+δhhR+δhhTδhhR+δvhTδvhR1+δvvT+δvvR+δvvTδvvR+δhvRδhvT.Furthermore, we assume |δij|=Δ and the phase of δij is uniformly distributed in [0,2π]. Using Taylor expansion and ignoring the second and higher order terms, we can get the approximation of the average bias of ZDR:(36)ZDRb¯≈10Δ,where •- means mathematical expectation. Since ZDRb has a symmetric distribution centered at 0, there is ZDRb¯=0. Hence, we use ZDRb¯ other than ZDRb¯. Figure 4 shows the relation between Δ and |ZDRb|¯. The red line is calculated from (36) while the blue line is obtained through Monte Carlo simulation in which δijT and δijR are generated from a random number generator and ZDRb is calculated from (35). In Figure 4 we see that the approximation from (36) agrees well with the result from Monte Carlo simulation.Figure 4
The relation betweenΔ and |ZDRb|¯.Using the same procedure, the average ICPR is derived in (37). Figure 5 shows the relation between Δ and ICPR¯:(37)ICPR¯≈20log⁡Δ.Figure 5
The relation betweenΔ and ICPR¯.From Figures4 and 5, we know that the calibration error δij has great impacts on ZDRb and ICPR. For a single spherical scatterer, to achieve ZDRb<0.1dB, Δ should be less than 0.01, which is really demanding for antenna pattern measurements. Moreover, we see that the relation between Δ and ZDRb¯ 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.Table 1
Array parameters.
Array size
64
×
64
Elements separation
λ
/
2
Sidelobe level ofFtay
-
40 dBStep 1.
Specify the polarization distortion calibration error|δij|.Step 2.
Generateαij, βij, and δij through a random number generator.Step 3.
CalculateT^ from (32) and R^ from (33).Step 4.
CalculateV^ from (17).Step 5.
CalculateZDRb from (20) and ICPR from (21).The simulation parameters are listed in Table2. U(a,b) means the uniform distribution in [a,b] and Arg(z) represents the phase of z. It should be pointed out that a -40dB 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 -54dB which is equal to that of the WSR-88D. Thus the simulated results with a -13dB uniform weighting and -40dB Taylor weighting are more comparable to those of radars with mechanical scanning antennas.Table 2
Simulation parameters.
Z
DR
0 dB
L
DR
-
∞ dB
|
δ
i
j
|
0.01
|
α
i
j
|
U
(
0,2
)
|
β
i
j
|
U
(
0,2
)
Arg(αij)
U
(
0,2
π
)
Arg(βij)
U
(
0,2
π
)
Arg(δij)
U
(
0,2
π
)The ranges ofαij and βij in Table 2 are determined based on the radiation pattern of a pair of crossed dipoles, which is written as(38)fdipoleθ,φ=cos⁡φ0cos⁡θsin⁡φsin⁡θ.The calibrated pattern f^dipole(θ,φ) is then written as(39)f^dipoleθ,φ=cos⁡φ0cos⁡θsin⁡φsin⁡θ·cos⁡φS0cos⁡θSsin⁡φSsin⁡θS-1≡1+εhhθ,φεhvθ,φεvhθ,φ1+εvvθ,φ.Choosing (θS,φS)=(60∘,45∘), we calculate εij(θ,φ). By using Matlab Curve Fitting Toolbox, we calculate the parameters αij, βij, and δij and show them in Table 3. R2 is called the coefficient of determination, which is a number that indicates how well data fit a statistical model. As shown in Table 3, R2 with respect to εij is 0.99, indicating a very good approximation performance of the linear error model. According to Table 3, we know αij,βij~U(0,2) is valid. Using the same procedure, we calculate the parameters αij, βij, and δij for a pair of crossed dipoles with the length of λ and show them in Table 4. Actually, the practical phased array usually has an element spacing of about λ/2. Thus the ranges of αij and βij should be better than the worst value in Table 4. In this paper, we assume |αij|<4 and |βij|<4.Table 3
The model parameters of a pair of crossed dipoles in the beam direction (60°, 45°).
α
β
δ
R
2
ε
hh
0.0000
-
1.1092
-
0.0013
0.99
ε
hv
0
0
0
None
ε
vh
1.1526
-
0.4989
0.0007
0.99
ε
vv
0.5769
0.0000
-
0.0013
0.99Table 4
The model parameters of a pair of crossed dipoles with the length ofλ in the beam direction (60°, 45°).
α
β
δ
R
2
ε
hh
-
1.0078
-
2.7556
0.0076
0.99
ε
hv
0
0
0
None
ε
vh
-
2.5394
-
0.3672
0.0003
0.99
ε
vv
2.3362
0.0000
-
0.0010
0.99Figures6 and 7 show the simulated |ZDRb|¯ and ICPR¯ in the beam scan area [60∘,90∘]×[0∘,45∘]. In Figure 6 most of |ZDRb|¯ 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 ZDRb is not obvious and ZDRb 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 -32.6dB at the broadside in Figure 7 is much larger than that of -40dB calculated from (37), indicating that the finite beamwidth considerably affects the measurement of LDR.Figure 6
|
Z
D
R
b
|
¯ in dB with |δij|=0.01, |αij|,|βij|~U(0,2).Figure 7
I
C
P
R
¯ in dB with |δij|=0.01, |αij|,|βij|~U(0,2).We then set|δij|=0.02, |αij|,|βij|∈U(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 |ZDRb|¯ 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 -26.5dB, increasing by about 7.5 dB compared with that calculated from the approximation of (37).Figure 8
|
Z
D
R
b
|
¯ in dB with |δij|=0.02, |αij|,|βij|~U(0,4).Figure 9
I
C
P
R
¯ in dB with |δij|=0.02, |αij|,|βij|~U(0,4).


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

With imperfectH/V channels, R^ and T^ can be written as(40)T^θ,φ=1+εhhTεhvTεvhT1+εvvTF^T,R^θ,φ=1+εhhRεhvRεvhR1+εvvRF^R.A and B are expressed as(41)A=1ηhvexp⁡jγhvηvhexp⁡jγvhηvvexp⁡jγvv,B=1τhvexp⁡jψhvτvhexp⁡jψvhτvvexp⁡jψvv.First, we analyze the case with a single spherical scatterer in the beam direction(θS,φS). Thus we just need to consider T^(θ,φ)θ=θS,φ=φS and R^(θ,φ)θ=θS,φ=φS. Since Xmn(θS,φS) and Ymn(θS,φS) compensate the phase displacement exp(jkr→mn·ar), we can get(42)T^θS,φS=1+δhhTδhvTδvhT1+δvvT·∑m=1M∑n=1NX^mnθS,φSAmn,R^θS,φS=1+δhhRδhvRδvhR1+δvvR·∑m=1M∑n=1NY^mnθS,φSBmn.As ∑m=1M∑n=1NX^mn(θS,φS)=1 and ∑m=1M∑n=1NY^mn(θS,φS)=1, the double summations on the right sides of (42) are close to the mathematical expectations of A and B. If γhv,γvh,ψhv,ψvh~U(0,2π), we have a¯hv=0, a¯vh=0, b¯hv=0, and b¯vh=0. In this situation, ZDRb can be calculated as(43)ZDRb=20log⁡1a¯vvb¯vv1+δhhT+δhhR+δhhTδhhR+δvhTδvhR1+δvvT+δvvR+δvvTδvvR+δhvTδhvR.If δijT=δijR=0, (43) reduces to(44)ZDRb=20log⁡1a¯vvb¯vv.Assuming |avv|=|bvv|<1, based on the definition of CIM in (5), (44) can be expressed as(45)ZDRb=2CIM.Figure10 shows the simulated |ZDRb|¯ of a single spherical scatterer based on (43), in which ηvv=τvv=0.99, γvv,ψvv~U(-10∘,10∘), ηhv=ηvh=τhv=τvh=0, |δij|=Δ, and Arg(δij)~U(0,2π). In Figure 10 when Δ<0.01, |ZDRb|¯ almost remains constant around 0.26 dB and when Δ>0.02, |ZDRb|¯ increases linearly with a slope less than 10. When Δ is small the channel imbalance has the main contribution to ZDRb. Otherwise, the polarization distortion calibration error is the dominant bias source.Figure 10
|
Z
D
R
b
|
¯ of a single spherical scatterer with imperfect H/V channels.In order to evaluate the bias under different conditions, we use the Monte Carlo simulation method again, which is shown below.Step 1.
Specify|δij|.Step 2.
Generateαij, βij, and δij from a random number generators and then calculate εij.Step 3.
GenerateAmn and Bmn for each element.Step 4.
CalculateR^ and T^.Step 5.
CalculateZDRb and ICPR.Figures11 and 12 show simulated |ZDRb|¯ and ICPR¯ in the beam direction (90∘,0∘), in which |αij|=|βij|=2, ηhv=ηvh=τhv=τvh=0, ηvv,τvv~N(0.99,0.012), and γvv,ψvv~U(-10∘,10∘). 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.Figure 11
|
Z
D
R
b
|
¯ of a single spherical scatterer and finite beamwidth.Figure 12
I
C
P
R
¯ of a single spherical scatterer and finite beamwidth.We then setηhv,ηvh,τhv,τvh~N(0.01,0.012), γhv,γvh,ψhv,ψvh~U(-10∘,10∘) and keep other parameters the same as those in Figures 11 and 12 to evaluate the impact of the channel isolation. The simulated |ZDRb|¯ and ICPR¯ considering the imperfect channel isolation are given in Figures 13 and 14 with the blue squares while the simulated |ZDRb|¯ and ICPR¯ from Figures 11 and 12 are still shown with the red triangles. In Figure 13 the impact of the imperfect channel isolation on ZDRb is not obvious. However, in Figure 14, the impact on ICPR is obvious when Δ is small.Figure 13
The impact of the channel coupling on|ZDRb|¯.Figure 14
The impact of the channel coupling onICPR¯.


## 3.1. Formulation

For a point target with the polarization scattering matrix (PSM)S in the direction (θ,φ) at the range r, the received voltage matrix can be written as(11)V=Cexp⁡-j2kr4πr2Rt·S·T·EATSRinc,where C is a gain term. Here the superscript “t” means matrix transpose. k=2π/λ and λ is the wavelength. EATSRinc is the unit excitation for H and V ports, which is written as(12)EATSRinc=1001.The received voltage matrix for distributed precipitations can be expressed as an integral. Consider(13)V=∫ΩRtθ,φ·Sθ,φ·Tθ,φdΩ,where Ω is the solid angle and dΩ=sin⁡θdθdφ. In (13), the gain term C and the term related to range r are dropped for the sake of simplicity. To retrieve S(θ,φ), the calibrated voltage matrix can be expressed as(14)V^=CR·V·CT=∫ΩCR·Rt·S·T·CTdΩ,where the calibration matrices CT and CR are expressed as(15)CT=Tθ,φθ=θS,φ=φS-1,CR=Rtθ,φθ=θS,φ=φS-1.CT and CR can be obtained through array pattern measurements. By defining(16)R^=R·CRt=R^hhR^hvR^vhR^vv,T^=T·CT=T^hhT^hvT^vhT^vv,the calibrated voltage matrix is written as(17)V^=∫ΩR^t·S·T^dΩ.Assuming(18)Sθ,φ=Shhθ,φ00Svvθ,φthe intrinsic differential reflectivityZDR is defined as(19)ZDR=10log⁡Shh2Svv2,where · means the ensemble average. The bias of ZDR can be calculated as(20)ZDRb=10log⁡PhhPvv-ZDR,where Pij (i,j=h,v) is the received power. Meanwhile, the integrated cross-polarization ratio (ICPR) is calculated as(21)ICPR=10log⁡PvhPhhwhich is the minimal linear depolarization ratio LDR that can be measured by a weather radar. After some trivial mathematical derivations, we get(22)ZDRb=10log⁡1+W1Zdr-1/2W2+W3Zdr1/2,ICPR=10log⁡W4+W5Zdr-1+W6Zdr-1/21+W1Zdr-1/2,where(23)ρhvexp⁡jϕDP=Shh∗SvvShh2Svv2,(24)W1=∫Ω2ReT^vhR^vhT^hh∗R^hh∗ρhvexp⁡jϕDPdΩ∫ΩThh2Rhh2dΩ,(25)W2=∫ΩT^vv2R^vv2dΩ∫ΩThh2Rhh2dΩ,(26)W3=∫Ω2ReT^vvR^vvT^hv∗R^hv∗ρhvexp⁡jϕDPdΩ∫ΩThh2Rhh2dΩ,(27)W4=∫ΩThh2Rhv2dΩ∫ΩThh2Rhh2dΩ,(28)W5=∫ΩTvh2Rvv2dΩ∫ΩThh2Rhh2dΩ,(29)W6=∫Ω2ReTvhRvvThh∗Rhv∗ρhvexp⁡jϕDPdΩ∫ΩThh2Rhh2dΩ.In (23) ρhv is called the copolar correlation coefficient and ϕDP represents the differential phase. The symbol ∗ means complex conjugate.According to (22), it is clear that ZDRb and ICPR are related to both the array patterns and the intrinsic Zdr. The impacts of Zdr and ρhvexp⁡(jϕDP) on ZDRb 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 Zdr=1 and ρhvexp⁡(jϕDP)=1 so that we can focus on the biases due to the radar system.


## 3.2. Array with Perfect H/V Channels

An array with perfectH/V channels means that the channel imbalance and isolation can be ignored. Hence, the transmission and reception patterns can be written as(30)Tθ,φ=faveθ,φ∑m=1M∑n=1Nexp⁡jkr→mn·arXmnθS,φS,Rθ,φ=faveθ,φ∑m=1M∑n=1Nexp⁡jkr→mn·arYmnθS,φS.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. SoR^ and T^ are written as(31)T^=T·CT=Funiθ,φfaveθ,φ·CT,R^=R·CRt=Ftayθ,φfaveθ,φ·CRt,where Funi(θ,φ) and Ftay(θ,φ) are the array factors of the uniform and Taylor weightings. According to the definitions of CT and CR, T^ and R^ can be modeled as(32)T^=F^uni1+εhhTθ,φεhvTθ,φεvhTθ,φ1+εvvTθ,φ,(33)R^=F^tay1+εhhRθ,φεhvRθ,φεvhRθ,φ1+εvvRθ,φ,where εij are the error terms after the calibration. F^uni and F^tay are the normalized array factors. The superscripts T and R represent the transmission and reception, which are usually dropped for simplicity. To simplify the analysis, εij is modeled as(34)εijθ,φ≈αijθ-θS+βijφ-φS+δij,where αij, βij, and δij are complex numbers. The unit of θ and φ is radian. If CT and CR have no error, there will be εij=0 at (θS,φS); that is, δij=0. However, due to the antenna pattern measurement errors, δij is not 0. In addition, αij and βij indicate the polarization variation near (θS,φS). 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 ofδij has the same level as the relative error upper bound Ef of the antenna measurements. Therefore, in the rest of this paper we just focus on δij.First, we analyze a simple case to get some insights towardsδij. We assume there is only one spherical scatterer in the beam direction (θS,φS), indicating ZDR=0dB and LDR=-∞ dB. So we can ignore the impacts of the finite beamwidth and sidelobes. Consequently, ZDRb can be calculated as(35)ZDRb=20log⁡1+δhhT+δhhR+δhhTδhhR+δvhTδvhR1+δvvT+δvvR+δvvTδvvR+δhvRδhvT.Furthermore, we assume |δij|=Δ and the phase of δij is uniformly distributed in [0,2π]. Using Taylor expansion and ignoring the second and higher order terms, we can get the approximation of the average bias of ZDR:(36)ZDRb¯≈10Δ,where •- means mathematical expectation. Since ZDRb has a symmetric distribution centered at 0, there is ZDRb¯=0. Hence, we use ZDRb¯ other than ZDRb¯. Figure 4 shows the relation between Δ and |ZDRb|¯. The red line is calculated from (36) while the blue line is obtained through Monte Carlo simulation in which δijT and δijR are generated from a random number generator and ZDRb is calculated from (35). In Figure 4 we see that the approximation from (36) agrees well with the result from Monte Carlo simulation.Figure 4
The relation betweenΔ and |ZDRb|¯.Using the same procedure, the average ICPR is derived in (37). Figure 5 shows the relation between Δ and ICPR¯:(37)ICPR¯≈20log⁡Δ.Figure 5
The relation betweenΔ and ICPR¯.From Figures4 and 5, we know that the calibration error δij has great impacts on ZDRb and ICPR. For a single spherical scatterer, to achieve ZDRb<0.1dB, Δ should be less than 0.01, which is really demanding for antenna pattern measurements. Moreover, we see that the relation between Δ and ZDRb¯ 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.Table 1
Array parameters.
Array size
64
×
64
Elements separation
λ
/
2
Sidelobe level ofFtay
-
40 dBStep 1.
Specify the polarization distortion calibration error|δij|.Step 2.
Generateαij, βij, and δij through a random number generator.Step 3.
CalculateT^ from (32) and R^ from (33).Step 4.
CalculateV^ from (17).Step 5.
CalculateZDRb from (20) and ICPR from (21).The simulation parameters are listed in Table2. U(a,b) means the uniform distribution in [a,b] and Arg(z) represents the phase of z. It should be pointed out that a -40dB 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 -54dB which is equal to that of the WSR-88D. Thus the simulated results with a -13dB uniform weighting and -40dB Taylor weighting are more comparable to those of radars with mechanical scanning antennas.Table 2
Simulation parameters.
Z
DR
0 dB
L
DR
-
∞ dB
|
δ
i
j
|
0.01
|
α
i
j
|
U
(
0,2
)
|
β
i
j
|
U
(
0,2
)
Arg(αij)
U
(
0,2
π
)
Arg(βij)
U
(
0,2
π
)
Arg(δij)
U
(
0,2
π
)The ranges ofαij and βij in Table 2 are determined based on the radiation pattern of a pair of crossed dipoles, which is written as(38)fdipoleθ,φ=cos⁡φ0cos⁡θsin⁡φsin⁡θ.The calibrated pattern f^dipole(θ,φ) is then written as(39)f^dipoleθ,φ=cos⁡φ0cos⁡θsin⁡φsin⁡θ·cos⁡φS0cos⁡θSsin⁡φSsin⁡θS-1≡1+εhhθ,φεhvθ,φεvhθ,φ1+εvvθ,φ.Choosing (θS,φS)=(60∘,45∘), we calculate εij(θ,φ). By using Matlab Curve Fitting Toolbox, we calculate the parameters αij, βij, and δij and show them in Table 3. R2 is called the coefficient of determination, which is a number that indicates how well data fit a statistical model. As shown in Table 3, R2 with respect to εij is 0.99, indicating a very good approximation performance of the linear error model. According to Table 3, we know αij,βij~U(0,2) is valid. Using the same procedure, we calculate the parameters αij, βij, and δij for a pair of crossed dipoles with the length of λ and show them in Table 4. Actually, the practical phased array usually has an element spacing of about λ/2. Thus the ranges of αij and βij should be better than the worst value in Table 4. In this paper, we assume |αij|<4 and |βij|<4.Table 3
The model parameters of a pair of crossed dipoles in the beam direction (60°, 45°).
α
β
δ
R
2
ε
hh
0.0000
-
1.1092
-
0.0013
0.99
ε
hv
0
0
0
None
ε
vh
1.1526
-
0.4989
0.0007
0.99
ε
vv
0.5769
0.0000
-
0.0013
0.99Table 4
The model parameters of a pair of crossed dipoles with the length ofλ in the beam direction (60°, 45°).
α
β
δ
R
2
ε
hh
-
1.0078
-
2.7556
0.0076
0.99
ε
hv
0
0
0
None
ε
vh
-
2.5394
-
0.3672
0.0003
0.99
ε
vv
2.3362
0.0000
-
0.0010
0.99Figures6 and 7 show the simulated |ZDRb|¯ and ICPR¯ in the beam scan area [60∘,90∘]×[0∘,45∘]. In Figure 6 most of |ZDRb|¯ 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 ZDRb is not obvious and ZDRb 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 -32.6dB at the broadside in Figure 7 is much larger than that of -40dB calculated from (37), indicating that the finite beamwidth considerably affects the measurement of LDR.Figure 6
|
Z
D
R
b
|
¯ in dB with |δij|=0.01, |αij|,|βij|~U(0,2).Figure 7
I
C
P
R
¯ in dB with |δij|=0.01, |αij|,|βij|~U(0,2).We then set|δij|=0.02, |αij|,|βij|∈U(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 |ZDRb|¯ 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 -26.5dB, increasing by about 7.5 dB compared with that calculated from the approximation of (37).Figure 8
|
Z
D
R
b
|
¯ in dB with |δij|=0.02, |αij|,|βij|~U(0,4).Figure 9
I
C
P
R
¯ in dB with |δij|=0.02, |αij|,|βij|~U(0,4).


## 3.3. Array with Imperfect H/V Channels

With imperfectH/V channels, R^ and T^ can be written as(40)T^θ,φ=1+εhhTεhvTεvhT1+εvvTF^T,R^θ,φ=1+εhhRεhvRεvhR1+εvvRF^R.A and B are expressed as(41)A=1ηhvexp⁡jγhvηvhexp⁡jγvhηvvexp⁡jγvv,B=1τhvexp⁡jψhvτvhexp⁡jψvhτvvexp⁡jψvv.First, we analyze the case with a single spherical scatterer in the beam direction(θS,φS). Thus we just need to consider T^(θ,φ)θ=θS,φ=φS and R^(θ,φ)θ=θS,φ=φS. Since Xmn(θS,φS) and Ymn(θS,φS) compensate the phase displacement exp(jkr→mn·ar), we can get(42)T^θS,φS=1+δhhTδhvTδvhT1+δvvT·∑m=1M∑n=1NX^mnθS,φSAmn,R^θS,φS=1+δhhRδhvRδvhR1+δvvR·∑m=1M∑n=1NY^mnθS,φSBmn.As ∑m=1M∑n=1NX^mn(θS,φS)=1 and ∑m=1M∑n=1NY^mn(θS,φS)=1, the double summations on the right sides of (42) are close to the mathematical expectations of A and B. If γhv,γvh,ψhv,ψvh~U(0,2π), we have a¯hv=0, a¯vh=0, b¯hv=0, and b¯vh=0. In this situation, ZDRb can be calculated as(43)ZDRb=20log⁡1a¯vvb¯vv1+δhhT+δhhR+δhhTδhhR+δvhTδvhR1+δvvT+δvvR+δvvTδvvR+δhvTδhvR.If δijT=δijR=0, (43) reduces to(44)ZDRb=20log⁡1a¯vvb¯vv.Assuming |avv|=|bvv|<1, based on the definition of CIM in (5), (44) can be expressed as(45)ZDRb=2CIM.Figure10 shows the simulated |ZDRb|¯ of a single spherical scatterer based on (43), in which ηvv=τvv=0.99, γvv,ψvv~U(-10∘,10∘), ηhv=ηvh=τhv=τvh=0, |δij|=Δ, and Arg(δij)~U(0,2π). In Figure 10 when Δ<0.01, |ZDRb|¯ almost remains constant around 0.26 dB and when Δ>0.02, |ZDRb|¯ increases linearly with a slope less than 10. When Δ is small the channel imbalance has the main contribution to ZDRb. Otherwise, the polarization distortion calibration error is the dominant bias source.Figure 10
|
Z
D
R
b
|
¯ of a single spherical scatterer with imperfect H/V channels.In order to evaluate the bias under different conditions, we use the Monte Carlo simulation method again, which is shown below.Step 1.
Specify|δij|.Step 2.
Generateαij, βij, and δij from a random number generators and then calculate εij.Step 3.
GenerateAmn and Bmn for each element.Step 4.
CalculateR^ and T^.Step 5.
CalculateZDRb and ICPR.Figures11 and 12 show simulated |ZDRb|¯ and ICPR¯ in the beam direction (90∘,0∘), in which |αij|=|βij|=2, ηhv=ηvh=τhv=τvh=0, ηvv,τvv~N(0.99,0.012), and γvv,ψvv~U(-10∘,10∘). 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.Figure 11
|
Z
D
R
b
|
¯ of a single spherical scatterer and finite beamwidth.Figure 12
I
C
P
R
¯ of a single spherical scatterer and finite beamwidth.We then setηhv,ηvh,τhv,τvh~N(0.01,0.012), γhv,γvh,ψhv,ψvh~U(-10∘,10∘) and keep other parameters the same as those in Figures 11 and 12 to evaluate the impact of the channel isolation. The simulated |ZDRb|¯ and ICPR¯ considering the imperfect channel isolation are given in Figures 13 and 14 with the blue squares while the simulated |ZDRb|¯ and ICPR¯ from Figures 11 and 12 are still shown with the red triangles. In Figure 13 the impact of the imperfect channel isolation on ZDRb is not obvious. However, in Figure 14, the impact on ICPR is obvious when Δ is small.Figure 13
The impact of the channel coupling on|ZDRb|¯.Figure 14
The impact of the channel coupling onICPR¯.


## 4. Array Analysis in STSR Mode

In the STSR mode, the received voltages for distributed precipitations are expressed as(46)VhtVvt=∫ΩRtθ,φ·Sθ,φ·Tθ,φ·shtsvtdΩ,where sh(t) and sv(t) represent two waveforms from the H and V channels. Assuming Shv(θ,φ)=Svh(θ,φ)=0, (46) can be written as(47)Vht=RhhShhThh+RvhSvvTvhsht+RhhShhThv+RvhSvvTvvsvt,Vvt=RhvShhThh+RvvSvvTvhsht+RhvShhThv+RvvSvvTvvsvt.As shown in (47), Vh(t) 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 ZDRb<0.1dB. 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 H and V channels can be written as(48)VhhtVhvtVvhtVvvt=Vht⊗s~htVht⊗s~vtVvt⊗s~htVvt⊗s~vt=∫ΩRt·S·T·sht⊗s~htsht⊗s~vtsvt⊗s~htsvt⊗s~vtdΩ,where s~h(t)=ghsh∗(t0-t) and s~v(t)=gvsv∗(t0-t) are the matched filters of the H and V channels, respectively. ⊗ means signal convolution. If sh(t) and sv(t) are completely orthogonal, we can get(49)sht⊗s~htsht⊗s~vtsvt⊗s~htsvt⊗s~vt∝1001.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,sh(t) and sv(t) cannot be completely orthogonal. Then we define(50)Pt=sht⊗s~htsht⊗s~vtsvt⊗s~htsvt⊗s~vt.So (48) can be written as(51)Vt=∫ΩRtθ,φ·Sθ,φ·Tθ,φ·PtdΩ.Accordingly, the calibrated voltage matrix V^(t) can be written as(52)V^t=CR·Vt·P-1t·CT,where CT and CR are defined in (15).Once the waveformssh(t) and sv(t) are known, P(t) 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 imperfectH/V 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 errorδij that has the same level as the relative error upper bound Ef of the antenna pattern measurement is critical for the biases of ZDR and LDR. δij should be under 0.01 to achieve ZDR<0.1dB, indicating that Ef should be under 1%. The imperfect H and V channels have considerable contributions to the biases of ZDR and LDR. 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 ZDRb<0.1dB.The finite beamwidth considerably affects the measurement ofLDR. However, the measurement of ZDR 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 LDR. Therefore, for a large array with a narrow beam that is commonly used for weather radars, the measurement performance of ZDR can be directly estimated through the measurement at the boresight, which can significantly simplify the analysis for the measurement of ZDR.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.



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*Source: 340704-2015-08-06.xml*