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  • View in gallery

    Two-way antenna patterns (45° plane). The solid line is the copolar pattern and the dashed line is the cross-polar pattern

  • View in gallery

    Two-way antenna patterns (45° plane). The cross-polar pattern is normalized to have the same maximum value as the copolar pattern

  • View in gallery

    Dependence of the cross-correlation coefficient |ρxh(0)| on Ldr for precipitation measurements

  • View in gallery

    A typical vertical profile of rain. Only the absolute calibration is applied

  • View in gallery

    Ldr profile of precipitation. The dashed line represents the uncalibrated Ldr and the solid line shows the calibrated Ldr. The point target calibration technique was used. Steady increase of Ldr values in the cloud region is caused by a decrease in SNR ratio

  • View in gallery

    A schematic graph showing the effect of the antennas on copolar (hh) and cross-polar (υh) measurements of a target with Ldr = −∞ dB. It should be noted that under the normal circumstances of atmospheric measurements where ShhSυυ the copolar measurements are mainly determined by the fthhfrhhShh since ftυhfrυhSυυ is at least two orders of magnitude smaller

  • View in gallery

    The magnitude of cross-correlation coefficient between cross- and copolar measurements as a function of Ldr. The thin solid line shows prediction of this behavior if no decorrelation compensation is applied. The thick solid line represents prediction obtained from the final calibration model. The o-shaped markers show the behavior of the correlation coefficient before calibration (same as Fig. 3) and the x-shaped markers show the behavior of the correlation coefficient |ρxh(0)| after the calibration

  • View in gallery

    Uncalibrated Ldr values of different precipitation measurements as a function of the calibrated Ldr. The solid line is given to facilitate the interpretation of the graph and shows a one-to-one relation between the calibrated and uncalibrated Ldr values

  • View in gallery

    Comparison between calibrated Ldr and uncalibrated Ldr values for a typical vertical precipitation profile. The improved polarimetric calibration procedure is used

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Improved Polarimetric Calibration for Atmospheric Radars

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  • 1 IRCTR, Delft University of Technology, Delft, Netherlands
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Abstract

Polarization properties of radar waves that are scattered from atmospheric objects are of great interest in meteorological studies. However, polarimetric radar measurements are often not sufficiently accurate for retrieving physical properties of targets. To compensate for errors, radar polarimetric calibration is applied. Typical calibrations are performed based on measurements of point targets with known scattering matrices located in the boresight of the antenna. Such calibration takes into account the polarization state of the antenna pattern only at one point. Since radar measurements of atmospheric phenomena involve distributed targets that fill the full antenna beam, point target radar calibrations are inadequate for meteorological studies.

This paper explains in detail the effects of the complete antenna patterns on weather echoes. It is shown that the conventional polarimetric calibration can be significantly improved by incorporating light-rain (<20 dBZ) zenith-pointing measurements into the calibration procedure. As a result, the sensitivity of cross-polar measurements can be improved by 7 dB on average. Also it is shown that the bias in co-cross-polar correlation coefficient can be reduced.

Current affiliation: DEOS, Delft University of Technology, Delft, Netherlands

Corresponding author address: Dr. Dmitri N. Moisseev, DEOS, Delft University of Technology, Thijsseweg 11, 2629 JA Delft, Netherlands. Email: d.moisseev@citg.tudelft.nl

Abstract

Polarization properties of radar waves that are scattered from atmospheric objects are of great interest in meteorological studies. However, polarimetric radar measurements are often not sufficiently accurate for retrieving physical properties of targets. To compensate for errors, radar polarimetric calibration is applied. Typical calibrations are performed based on measurements of point targets with known scattering matrices located in the boresight of the antenna. Such calibration takes into account the polarization state of the antenna pattern only at one point. Since radar measurements of atmospheric phenomena involve distributed targets that fill the full antenna beam, point target radar calibrations are inadequate for meteorological studies.

This paper explains in detail the effects of the complete antenna patterns on weather echoes. It is shown that the conventional polarimetric calibration can be significantly improved by incorporating light-rain (<20 dBZ) zenith-pointing measurements into the calibration procedure. As a result, the sensitivity of cross-polar measurements can be improved by 7 dB on average. Also it is shown that the bias in co-cross-polar correlation coefficient can be reduced.

Current affiliation: DEOS, Delft University of Technology, Delft, Netherlands

Corresponding author address: Dr. Dmitri N. Moisseev, DEOS, Delft University of Technology, Thijsseweg 11, 2629 JA Delft, Netherlands. Email: d.moisseev@citg.tudelft.nl

1. Introduction

During recent years there has been a growing interest in the use of radar polarimetry for atmospheric studies. This has resulted in the use of various polarimetric parameters for the characterization of the meteorological media (Zrnić and Ryzhkov 1999). However, little attention has been given to the polarimetric calibration of atmospheric radars. Current calibration techniques are based on the measurement of known point targets. This approach is based on the assumption that the polarimetric measurements with the radar can be described by the polarimetric properties of only one point of the radar antenna pattern. However, atmospheric radars most of the time are used for studies of extended radar objects and thus full antenna pattern information must be taken into account.

The main goal of using calibration methods based on point-target measurements is to estimate the distortion matrices (McCormick 1981), which characterize the transmit and receive channels of the system. Many different approaches are described in the literature to retrieve these distortion matrices. The main difference between them lies in the assumptions which are used in order to simplify the calibration procedure. It was shown by Sarabandi and Ulaby (1990) that if the antenna system and two orthogonal channels of the radar system are modeled as a four-port passive network, the calibration can be simplified to the measurement of a metallic sphere. Another approach is to assume a good polarization isolation of the system (>25 dB); then the polarimetric calibration can be done by using a metallic sphere and an unknown highly depolarizing target (Sarabandi et al. 1990). The use of the assumption of reciprocity of the antenna system was demonstrated by Unal et al. (1994), and because of this assumption the polarimetric calibration can be carried out by using a single rotatable dihedral corner reflector. The use of weather echoes for calibration was demonstrated by Hubbert and Bringi (2001). However, in this paper the point target calibration formalism was also employed as the basis for the calibration procedure.

In this article we analyze the errors that are implied by the point target calibration formalism. Based on this analysis a correction method for existing calibration procedures is proposed. It is demonstrated that by applying this correction we can obtain more accurate cross-polar measurements. This new polarimetric calibration procedure is demonstrated using a set of precipitation measurements collected with the Delft Atmospheric Research Radar (DARR). The DARR description is given by Ligthart and Nieuwkerk (1980).

2. Point target polarimetric calibration

For an ideal polarimetric system, the measured scattering matrix of an object, 𝗠, is identical to the actual scattering matrix, 𝗦. However, in real radar systems these matrices are different. Usually this difference is explained by the divergence of the expected transmit and receive polarization states from the actual ones. For example, a radar transmit (or receive) basis that is treated as a linear hυ basis is in reality a nonorthogonal basis with polarization vectors that are slightly elliptical. In this case the measured scattering matrix can be represented as
where 𝗥 and 𝗧 are the antenna distortion receive and transmit 2 × 2 matrices, or, in other words, the matrices that transform the desired polarimetric basis into the actual antenna basis. This equation provides the basis for the point target polarimetric radar calibration. It can be seen that if no assumptions are made about the 𝗥 and 𝗧 matrices, three known point targets are required to retrieve all elements of the distortion matrices.
Following the assumption used by Unal et al. (1994) and Hubbert and Bringi (2001) that the radar can be considered to be reciprocal (𝗠T = 𝗠), the formulation can be simplified as
T
where
i1520-0426-19-12-1968-e3
and δ1, δ2 represent coupling terms between the radar polarization channels, f is the one-way copolar channel imbalance, and T11 is the coefficient that combines the absolute calibration constant and the radio wave propagation coefficient. It should be noted that since we are interested only in relative polarimetric calibration, the coefficient T11 is not considered in our discussion for the sake of simplicity.
Using the formulation (2) and (3) and taking into account that the matrices 𝗠 and 𝗦 are symmetric, we can rewrite (2) as
i1520-0426-19-12-1968-e4
where s and m are vector representations of the matrices 𝗠 and 𝗦,
i1520-0426-19-12-1968-eq1
It is common to use the polarimetric covariance matrix 𝗖 = 〈s · s+〉 for the description of atmospheric targets (Santalla et al. 1999; Ryzhkov 2001); here + is the Hermitian transpose operator and 〈 〉 denotes an ensemble average. Using a general relationship
i1520-0426-19-12-1968-eq2
where 𝗔, 𝗕, 𝗖, and 𝗗 are n × n matrices, and a, c, are n2 vector representations of 𝗔 and 𝗖 formed by stacking the rows of the matrices, the calibrated covariance matrix 𝗖 can be obtained from (4) as follows:
cmcs
where cm and cs are the vector representations of the measured and actual object's covariance matrices and 𝗗 is the system distortion matrix, which is defined as
i1520-0426-19-12-1968-e6
Here ⊗ denotes the Kronecker matrix product and * denotes complex conjugation.

Assuming we know f, δ1, and δ2, one can invert (5) to obtain the calibrated covariance matrix of a radar object. However, this approach is based on the assumption that it suffices to estimate the distortion matrix at only one point of the antenna pattern. This approach can be erroneous, especially for atmospheric studies, where the complete antenna pattern plays a role in the radar measurement.

3. Effect of the complete antenna patterns on atmospheric measurements

a. Polarimetric measurements of distributed targets

For a distributed target measurement, the measured scattering matrix can be related to the target scattering matrix by (Blanchard and Jean 1983; Doviak and Zrnić 1993):
i1520-0426-19-12-1968-e7
where
i1520-0426-19-12-1968-eq2a
is the receive (or transmit) distortion matrix whose elements are antenna patterns for all polarization states and (θ, ϕ) is the scattering matrix of an extended radar target for the azimuth angle θ and the elevation angle ϕ. This formulation gives the most general description of the influence of the radar system on polarimetric measurements. From (7) it follows that in order to perform an accurate polarimetric radar calibration, one requires knowledge of eight 2D antenna patterns (four if the reciprocity assumption of the antenna system is used). However, measuring antenna patterns accurately is complicated and time consuming, and not an option for most atmospheric radars. That is why it is important to examine how the antenna patterns influence the polarimetric measurements of atmospheric objects and what can be done to reduce this effect.

b. Copolar channel imbalance

The channel imbalance f is an important parameter; it can be seen from (2) and (3) that for radars with good polarization isolation (small δ1, δ2) the accuracy of the measured differential reflectivity (Zdr = 〈|Mhh|2〉/〈|Mυυ|2〉) is directly related to f.

In order to determine how the complete antenna pattern influences Zdr measurements, we used the antenna patterns for DARR (see Fig. 1). Coherent measurements of the antenna patterns were available in principal and 45° planes (Aubry and Zijderveld 1999). The effects of the antenna patterns were best illustrated using 45° plane measurements, in conjunction with the assumption of rotational symmetry. Using (7) we have estimated the measured Zdr value for a target with Zdr = 0 dB. Based on these calculations we have found that the difference between copolar polarization channels due to the use of full antenna patterns is small and gives an uncertainty in Zdr measurements of about 10−2 dB.

Due to uncertainties in the antenna pattern model and in the antenna measurements, especially for far sidelobes, this estimation does not give us the exact value of the influence of the complete antenna patterns on f. However, it does give us an estimate of the order of magnitude of this influence. Thus for most atmospheric studies the required Zdr accuracy of about 0.1 dB can be achieved by applying a point target polarimetric calibration.

c. Polarization isolation for a distributed target

The other important characteristic of an antenna system is the polarization isolation. The polarization isolation defines how pure the polarization state of a radar is. For atmospheric studies the polarization isolation of the radar determines the minimum possible measured linear depolarization ratio values (Ldr = 〈|Mυh|2〉/〈|Mhh|2〉). If the point calibration notation is used, the polarization isolation ρa is given by
i1520-0426-19-12-1968-e8
where 〈|Mhh|2〉 and 〈|Mυh|2〉 are averaged measured powers of hh, and υh elements of the scattering matrix of a polarimetrically isotropic nondepolarizing target (〈|Sυh|2〉 = 0, 〈|Shh|2〉 = 〈|Sυυ|2〉). It can be seen that the smallest measurable Ldr value is fully determined by the antenna isolation value, namely Ldrmin = 1/ρa. Based on the point target calibration (Unal et al. 1994) it was determined that this value is 33 dB for DARR.

However, many atmospheric measurements with DARR showed that this value is never achieved and the minimum measured Ldr value is only −27.5 dB. This Ldr value does not represent the characteristic of atmospheric echoes, but the antenna polarization isolation. This difference between the predicted and measured polarization isolation was first described by Blanchard and Jean (1983). It was shown that this discrepancy is explained by the way in which the polarization isolation is determined: whether it is estimated from single point (boresight) measurements or by taking into account the complete antenna patterns.

From (7) and (8), we can define the polarization isolation for an extended target as
i1520-0426-19-12-1968-e9
where Ω is the solid angle and dΩ = sinθdθdϕ. Using (9) we estimated the polarization isolation for DARR; it was found to be about 27 dB. This value of the polarization isolation is in good agreement with the atmospheric measurements.

d. Effect of the antenna patterns on the copolar and cross-polar correlation

The measurements of any element of the scattering matrix are influenced by the co- and cross-polar antenna patterns (Fig. 1). These antenna patterns determine the spatial structure of the copolar and cross-polar channel responses and are represented in the distortion matrices as fthh, ftυυ, frhh, frυυ for copolar patterns and fthυ, ftυh, frhυ, frυh for the cross-polar ones. In the cross-polar measurement of weakly depolarizing targets (most atmospheric objects), the influence of the cross-polar channel response is strong. Compensation for this influence is an important part of any calibration procedure. In a point target calibration procedure, the copolar and cross-polar channel responses are assumed to have no spatial dependences. However, in reality, this can be an erroneous assumption. From Fig. 1 and in more detail from Fig. 2 it can be seen that the co- and cross-polar antenna patterns have different shapes and as a result the observed volumes are also different.

The effect of the difference between these antenna patterns can clearly be seen in the observations of the magnitude of cross-correlation coefficient |ρxh(0)| = |〈MhhM*υh〉/〈|Mhh|2〉〈|Mυh|2| (or |ρ(0)|) between copolar and cross-polar measurements of atmospheric objects. Figure 3 shows the dependence between the cross-correlation coefficient and the measured Ldr values. This figure presents data from six different measurements of light rain, in five cases with the radar pointed to the zenith and in one case at an 80° elevation angle. The figure shows measurement corresponding not only to the light rain but also to the melting layer and to the return from above the melting layer. In the light rain region, the measurements with Ldr ≈ −27 dB had a signal-to-noise ratio for cross-polar measurements of about 15 dB. More details about the signal processing used to calculate the correlation coefficient can be found in section 4b.

It is commonly assumed that the correlation coefficient between pure copolar and cross-polar measurements of atmospheric objects is zero (Antar and Hendry 1985). Although recent studies by Ryzhkov (2001) have shown that in some cases the correlation coefficient can be nonzero, we do not expect that this effect will be significant for the zenith measurements of precipitation. So if this assumption is used, the further behavior of the scatterplot is clear and shows that with the increase of the Ldr values the correlation coefficient should decrease. However, the value of the cross-correlation coefficient for Ldr values close to −27 dB requires more explanation, as follows.

While the copolar measurements are mainly determined by the copolar response of the target, the cross-polar measurements are strongly affected by coupling between the polarization channels. Thus, if the measurements of the target are performed with Ldr = −∞ dB, the expected value of |ρxh(0)| from the point of view of point target calibration will be 1, since it does not have any spatial dependence and the measured cross-polar values are determined purely by the coupling from the copolar channel. But if the complete antenna patterns are taken into account, this value will be defined by the mismatch between the co- and cross-polar antenna patterns. In this case the co- and cross-polar measurements correspond to different radar volumes, and this results in the decorrelation between these two measurements.

To check whether this effect is strong enough to explain the observed values of the cross-correlation coefficient, we used the measured antenna patterns to calculate the measured |ρxh(0)| for the case in which the copolar and cross-polar returns are completely correlated. To do this, we again used the 45° plane measurements and the assumption of rotational symmetry, as in section 3b. While this assumption is not generally valid for the cross-polar antenna patterns, it can be used for the purpose of this article, as shown below.

The two-dimensional distribution of the gain function of the cross-polar response of the antenna has four clear maxima: two in the 45° plane (see Fig. 2) and two in the 135° plane; for the other angles the cross-polar gain becomes smaller, reaching its minimum in the main planes. As a result, if we used rotationally symmetric patterns obtained from the 45° plane antenna pattern, the intersection between the radar volumes for the copolar and cross-polar measurements is larger than in reality; hence we overestimate the correlation between the copolar and the cross-polar reflections. If the estimated magnitude of the correlation coefficient is significantly smaller than unity it will prove our hypothesis.

Let us calculate the correlation coefficient between cross-polar and copolar signals. By using (7), the received average power of copolar reflections can be written as
i1520-0426-19-12-1968-e10
We should note that due to the fact that reflections from different volumes are uncorrelated, we have that 〈hh(θ, ϕ)*hh(θ′, ϕ′)〉 = 2σ2Ω · δ(θθ′, ϕϕ′), where 2σ2Ω is the power per unit solid angle. Then, assuming uniform beam filling, (10) can be rewritten as
i1520-0426-19-12-1968-e11
For the cross-polar averaged power we have
i1520-0426-19-12-1968-e12
To obtain (12) we have assumed that 〈|Shh|2〉 = 〈|Sυυ|2〉 and (〈ShhS*υυ〉)/(〈|Shh|2〉〈|Sυυ|2) = 1. Finally the covariance is
i1520-0426-19-12-1968-e13

Using (11), (12), and (13) we calculated the cross-correlation coefficient and found it to be equal to 0.74. As expected, this value is larger than the measured value of 0.5 (see Fig. 3), but it proves that mismatching of the antenna patterns can have a significant influence on |ρxh(0)|.

Based on the above-presented studies we can conclude that

  • for the Zdr measurements, a common point target calibration can be used,
  • polarization isolation and thus the coupling terms (δ1 and δ2) are strongly dependent on the complete antenna pattern.

4. Improvement of the polarimetric radar calibration procedure

a. Use of light-rain zenith measurements for improvement of the calibration

The results presented above show that the polarimetric radar calibration procedure based on point target measurements does not give an adequate description of the influence of the radar antennas on cross-polar properties of atmospheric echoes. It also indicates that if a point target calibration procedure is combined with the measurements of a known distributed target a much better description of the effect of the radar system on polarimetric measurements can be achieved. However, the main problem of this approach is the availability of a distributed target with known polarimetric characteristics.

The most obvious candidate for the role of a known distributed target is light rain measured with the radar pointing to the zenith. Generally we would expect that for this type of measurement the light rain will behave as a polarimetrically isotropic nondepolarizing target, meaning that the actual Ldr of light rain is much smaller than the antenna limit. However, this assumption can be violated in some special cases. It was shown by Ryzhkov et al. (1999) that hydrometeors can have a nonzero mean canting angle, which will result in nonzero Ldr values even for a vertically pointing radar. This is usually caused by wind shear or updraft effects. Another effect (Jameson and Durden 1996) that can contribute to the failure of our assumption is that large raindrops can oscillate (change their shape). The oscillation will result in an increase of the Ldr values. These two effects can restrict the use of light rain for calibration. A reduction of these effects is feasible, however.

The oscillation of raindrops as well as a nonzero canting angle mainly contribute to the changes of polarimetric properties of large raindrops. This can be efficiently reduced by applying Doppler processing. Furthermore, these effects were mainly seen for strong rain intensities (reflectivity above 30 dBZ), while we use only light-rain data with values of reflectivities not larger than 20 dBZ (Fig. 4).

b. Signal processing used for the estimation of the polarimetric properties of rain

Several signal processing algorithms were applied to reduce the influence of the following four effects:

  • ground clutter,
  • noise,
  • change of the polarimetric properties of large raindrops, and
  • staggered polarimetric measurements.

Since very accurate measurements of the small cross-polar signal are paramount for the calibration procedure the appropriate clutter suppression should be applied even to 90° elevation measurements of rain. The clutter suppression technique employed here is based on the use of the copolar correlation coefficient (Ryzhkov and Zrnić 1998; Moisseev et al. 2000) calculated for every Doppler resolution cell. For ground clutter these values are generally smaller than 0.7 and for atmospheric objects they are higher than 0.8. Thus, the Doppler cells that are contaminated by ground clutter can be successfully identified and suppressed. We should note that this processing also rejected cells with a poor signal to noise ratio (SNR). For further reduction of the noise, the averaged noise power was subtracted from all power measurements. Moreover, for all measurements presented in this paper averaging over 40 s was used.

In Fig. 4 the averaged vertical profiles of precipitation are given. It should be noted that in the cloud above the melting layer the signal-to-noise ratio decreases; as a result the SNR is almost unity for cross-polar measurements at the height of 4.5 km. This effect also causes a steady increase of Ldr values above the melting layer in Fig. 5. Therefore, for our analysis only data in the rain and melting layer were used, where SNR is better than 15 dB for the cross-polar measurements.

The possibility of large distorted drops affecting the processing was reduced by examining the behavior of Zdr and Ldr as functions of Doppler velocity (Unal et al. 2001). As a result, Doppler spectra, where changes in these parameters for different Doppler resolution cells are observed, were discarded.

Since DARR has only one receiver channel all covariances in the measured covariance matrix, 𝗖m, are obtained at nonzero time lag. Thus, it is necessary to estimate the values of this covariances at zero time lag. It can be shown that the difference between the cross-correlation calculated at zero time lag and at the time lag τ is an extra phase component ωτ in the cross spectrum (Unal and Moisseev 2002). Thus, by compensating this phase difference on a calculated cross spectrum and performing inverse FFT we can obtain a very accurate estimate of the covariance at zero time lag [for more information see Unal and Moisseev (2002)]. Applying this step to all covariances in 𝗖m we obtain a covariance matrix of precipitation as if all elements of the scattering matrix were measured simultaneously.

c. Improved calibration procedure and calibration results

From the above we can conclude that the point target calibration assumes that the antenna polarization states are completely polarized and thus they can be fully characterized by two 2 × 2 matrices, 𝗧 and 𝗥. In particular, it has been shown that in the case of distributed target measurements the correlation coefficient between direct and coupling terms of antenna polarization states is not equal to unity, and thus the assumption that antenna polarization states are completely polarized is not valid. Thus, in order to describe such partially polarized states the combined second-order distortion matrix 𝗗 should be used. Moreover, it should be noted that this distortion matrix cannot be determined from 𝗧 and 𝗥 alone, but needs to be corrected based on actual measurements of an extended target to take into account the full antenna pattern. In this section the procedure to improve calibration results for atmospheric studies based on measurements of light rain added to the results of the point target calibration is discussed.

As is shown above, the channel imbalance obtained from the point target calibration is precise enough to retrieve the Zdr values. On the other hand, the cross-polar measurements are very sensitive to the calibration procedure and thus the polarimetric calibration should be modified in order to improve the accuracy of these measurements. Based on this, we can postulate that the channel imbalance f can be retrieved from the point target calibration while the light-rain measurements can be used to improve the cross-polar measurements.

To understand how to correct for the cross-polar errors, one needs to examine how Myx and Mxx are affected by antenna pattern terms. For a target having Ldr = −∞ dB, the measured voltage will be given as
i1520-0426-19-12-1968-e14
where x, y = h, υ (see Fig. 6 for an example of Mυh measurement). The copolar measurements, Mxx, of such a target will be defined as
i1520-0426-19-12-1968-e15
(see Fig. 6). Since for atmospheric echoes we do not expect Zdr to exceed 10 dB, meaning that ShhSυυ, and in the case of having a radar with a good polarization isolation (>20 dB) we can simplify the expression (15) by neglecting the term ftyxfryxSyy and thus this expression can be rewritten as
i1520-0426-19-12-1968-eq3
If we now will consider the covariance 〈MxxM*yx〉 we see that it is defined by the sum of the integrals with the elements under the integral sign that involve terms that are the first-order in fxy(θ, ϕ) [or f*xy(θ, ϕ)]. It should be noted that these elements are mainly responsible for the decorrelation between co- and cross-polar measurements. Of course there are other elements of the distortion matrix that include higher orders of the coupling terms and influence this covariance, but as we have shown above these elements can be neglected. In terms of the point target calibration the elements of the distortion matrix that are causing decorrelation are those that are first-order in δi (or δ*i), where i = 1, 2. Under the current assumption of reciprocity of the radar system, the co-cross-polar covariances should be equal for the measurements of light rain; this also is confirmed by our measurements. Therefore, the decorrelation between copolar and cross-polar measurements due to the antenna pattern is characterized by one correlation coefficient |ρxh(0)|. In order to take this decorrelation into account one should multiply the elements of the matrix 𝗗 that are first-order in δi (or δ*i) by the cross-correlation coefficient |ρxh(0)| obtained from the light-rain measurements. As a result we can obtain the modified distortion matrix, , as
i1520-0426-19-12-1968-e16
where .* denotes the operator of element by element matrix multiplication.

A second step needed to improve the polarimetric calibration is to compensate for the difference between the polarization isolation estimations, (8) and (9). At first glance one can conclude that this can be done by solving (5) for δ1 and δ2. However, this solution is sensitive to the accuracy of f and it depends on the decorrelation compensation. Although the estimation of f from the point target calibration is accurate enough for the Zdr measurements, the order of magnitude of its error is comparable to the magnitudes of coupling terms δ1 and δ2. This means that the accurate solution of (5) for δ1 and δ2 is impossible. However, the polarization isolation estimates can be corrected simply by multiplying δ1 and δ2 by ρ̃a/ρa, where ρ̃a is the complete antenna polarization isolation and is equal to 1/Ldr from the light-rain measurements, and ρa is the point target isolation from (8). By doing so we keep the phases of δ1 and δ2 and the ratio δ1/δ2 as determined from point target calibration measurements and we adjust the amplitudes of the coupling terms to include the complete antenna pattern gain distribution. It should be noted that this step of the correction applies to each factor of δ1 and δ2 in the distortion matrix, , and not just to the terms that are the first-orders in δ1 and δ2. It is most simply implemented by correcting the δi terms (i = 1, 2) in the transmission matrix 𝗧 of Eq. (3), or in the 3 × 3 matrix of Eq. (4), from which D is determined.

These two simple procedures allow us to compensate for some discrepancy between the point target calibration and the measurements of extended targets. Of course, the assumptions underlying the current polarimetric calibration might not be valid for all cases; however, the use of these assumptions allows us to improve the polarimetric calibration method significantly. Moreover, these assumptions are less strong than the ones used when the point target calibration is applied, and thus it is implied that polarimetric characteristics of the antenna are solely defined by the polarimetric properties at the boresight direction.

In total six precipitation measurements were used to demonstrate the performance of the improved polarimetric calibration method. Two measurements were carried out on 27 November 1997 and the other four on 23 December 1997. Five of these measurements were obtained with the radar pointing to the zenith and one with a radar elevation angle of 80°.

In Fig. 5 one of the zenith measurements is used to show the result of the point target calibration procedure. It can be seen that the Ldr values have hardly changed after the calibration.

The scatterplot of the cross-correlation coefficient between cross- and copolar measurements as a function of the Ldr values obtained for all the measurements is shown in Fig. 7. Also two curves that represent the predicted behavior of the correlation coefficient are shown; the thin line shows the predicted behavior of |ρxh(0)| if no decorrelation compensation is applied to the estimate of the combined distortion matrix and the thick line shows the prediction that is obtained when the decorrelation compensation (16) is applied. These two lines are obtained by applying the combined distortion matrices, 𝗗 and , with and without decorrelation compensation, to the modeled covariance matrix of precipitation, as
i1520-0426-19-12-1968-eq4
where the covariance matrix, 𝗖s, is given as follows:
i1520-0426-19-12-1968-e17
and the Ldr values vary from −60 to −5 dB. The calibration parameters that were used to calculate 𝗗 and are given in Table 1, it should be noted that in both cases the second step of the calibration correction is applied, namely the δi terms (i = 1, 2) are multiplied by ρ̃a/ρa. As we have discussed earlier the expected value of the co-cross-polar correlation coefficient for atmospheric echoes is zero; this is also is assumed in (17). Moreover, we also have shown that the bias of the measured correlation coefficient is caused by the imperfections of the radar. Naturally any calibration procedure should predict the behavior of the measurands. It can be seen, however, that if no decorrelation compensation is used, the prediction, calculated from cm = 𝗗cs and (17), gives wrong estimates of the measured correlation coefficient. On the other hand, the modified distortion matrix gives the right estimate of the uncalibrated correlation coefficient behavior and follows the scatterplot of the measured uncalibrated |ρxh(0)|, which is depicted by o-shaped markers, as shown in Fig. 7. Moreover, if we estimate the co-cross-polar correlation coefficient from our measurements by applying the new calibration procedure,1 by computing cs = −1cm, we obtain a less biased estimate of the correlation coefficient, as shown by x-shaped markers. The other important conclusion that we can draw from this figure is that if the co-cross-polar correlation coefficient is used for the description of the media, extreme care should be taken. As we can see, even for Ldr values as high as −15 dB (Fig. 7) the correlation coefficient |ρxh(0)| has a nonnegligible bias.

In Fig. 8 the scatterplot of the resulting Ldr values for all six precipitation measurements is shown. This figure shows that the proposed calibration procedure gives an improvement of about 7 dB in antenna isolation. It also shows that uncalibrated Ldr values have a bias of more than 1 dB for all measurements of the targets with the actual Ldr values of less than −20 dB. The result of the new polarimetric calibration is also demonstrated on the single precipitation profile, measured with the radar pointing to the zenith, as shown in Fig. 9. We can see that the calibration has a nonnegligible effect on rain measurements. Moreover, we also can observe that the measurements of the melting layer are hardly influenced by the procedure. That is expected since the melting layer is a depolarizing target with Ldr values of about −15 dB and as is shown in Fig. 8 the effect of the calibration on Ldr estimations of such targets is small.

5. Summary and conclusions

In this article the effect of complete antenna patterns on the polarimetric measurements of atmospheric phenomena was analyzed. This study is based on the use of the measured antenna patterns as well as on the analysis of the precipitation measurements with DARR. The goal of this paper was to investigate the sensitivity of the polarimetric calibration to complete antenna patterns for atmospheric radars. It was shown that the common polarimetric calibration techniques, which imply that the polarimetric imperfection of a radar can be described from the measurements of known targets located at the boresight direction of the antenna, are not valid for the polarimetric measurements of atmospheric objects. The limitation of this assumption can clearly be seen in cross-polar measurements. It was also demonstrated that due to the antenna effects, both the Ldr values and the co-cross-polar correlation coefficient |ρxh(0)| values are biased. This means that due to imperfection of the polarization channels, below about −20 dB the Ldr measurements have a nonnegligible bias for DARR. The co-cross-polar correlation coefficient has a bias of more than 0.1 even for the melting-layer measurements with Ldr values of −15 dB.

Based on this study a new polarimetric radar calibration procedure is proposed. This new calibration method uses light-rain measurements in order to take into account the extended nature of weather echoes. This combination of a point target calibration method and light-rain measurements has led to a substantial improvement in the cross-polar measurements of atmospheric objects. This improvement mainly manifests itself in an increase of sensitivity of the cross-polar measurements by more than 7 dB and in a reduction of the effects of the antenna on the co-cross-polar correlation coefficient |ρxh(0)| measurements.

It should be noted, however, that the obtained data should carefully be analyzed; this is needed because polarimetric measurements are affected not only by an imperfection of the polarization channels, but also by noise, variance of the estimates of a covariance matrix elements, time changes of radar system characteristics, etc. All these effects can influence the performance of the polarimetric calibration and should be the topic of a future research. Therefore in our opinion the proposed calibration procedure should mainly be used to evaluate errors that are introduced by a system to polarimetric measurements and not to make attempts to measure Ldr below the antenna limit. Another use of this calibration procedure is to decrease the bias in the Ldr and |ρxh(0)| estimates, which, as have been shown, is rather large even for the polarimetric measurements above the system limit.

Acknowledgments

The authors wish to thank the anonymous reviewers for their valuable comments. Especially the authors would like to acknowledge the contribution of one of the reviewers whose comments have greatly helped to clarify the manuscript.

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Fig. 1.
Fig. 1.

Two-way antenna patterns (45° plane). The solid line is the copolar pattern and the dashed line is the cross-polar pattern

Citation: Journal of Atmospheric and Oceanic Technology 19, 12; 10.1175/1520-0426(2002)019<1968:IPCFAR>2.0.CO;2

Fig. 2.
Fig. 2.

Two-way antenna patterns (45° plane). The cross-polar pattern is normalized to have the same maximum value as the copolar pattern

Citation: Journal of Atmospheric and Oceanic Technology 19, 12; 10.1175/1520-0426(2002)019<1968:IPCFAR>2.0.CO;2

Fig. 3.
Fig. 3.

Dependence of the cross-correlation coefficient |ρxh(0)| on Ldr for precipitation measurements

Citation: Journal of Atmospheric and Oceanic Technology 19, 12; 10.1175/1520-0426(2002)019<1968:IPCFAR>2.0.CO;2

Fig. 4.
Fig. 4.

A typical vertical profile of rain. Only the absolute calibration is applied

Citation: Journal of Atmospheric and Oceanic Technology 19, 12; 10.1175/1520-0426(2002)019<1968:IPCFAR>2.0.CO;2

Fig. 5.
Fig. 5.

Ldr profile of precipitation. The dashed line represents the uncalibrated Ldr and the solid line shows the calibrated Ldr. The point target calibration technique was used. Steady increase of Ldr values in the cloud region is caused by a decrease in SNR ratio

Citation: Journal of Atmospheric and Oceanic Technology 19, 12; 10.1175/1520-0426(2002)019<1968:IPCFAR>2.0.CO;2

Fig. 6.
Fig. 6.

A schematic graph showing the effect of the antennas on copolar (hh) and cross-polar (υh) measurements of a target with Ldr = −∞ dB. It should be noted that under the normal circumstances of atmospheric measurements where ShhSυυ the copolar measurements are mainly determined by the fthhfrhhShh since ftυhfrυhSυυ is at least two orders of magnitude smaller

Citation: Journal of Atmospheric and Oceanic Technology 19, 12; 10.1175/1520-0426(2002)019<1968:IPCFAR>2.0.CO;2

Fig. 7.
Fig. 7.

The magnitude of cross-correlation coefficient between cross- and copolar measurements as a function of Ldr. The thin solid line shows prediction of this behavior if no decorrelation compensation is applied. The thick solid line represents prediction obtained from the final calibration model. The o-shaped markers show the behavior of the correlation coefficient before calibration (same as Fig. 3) and the x-shaped markers show the behavior of the correlation coefficient |ρxh(0)| after the calibration

Citation: Journal of Atmospheric and Oceanic Technology 19, 12; 10.1175/1520-0426(2002)019<1968:IPCFAR>2.0.CO;2

Fig. 8.
Fig. 8.

Uncalibrated Ldr values of different precipitation measurements as a function of the calibrated Ldr. The solid line is given to facilitate the interpretation of the graph and shows a one-to-one relation between the calibrated and uncalibrated Ldr values

Citation: Journal of Atmospheric and Oceanic Technology 19, 12; 10.1175/1520-0426(2002)019<1968:IPCFAR>2.0.CO;2

Fig. 9.
Fig. 9.

Comparison between calibrated Ldr and uncalibrated Ldr values for a typical vertical precipitation profile. The improved polarimetric calibration procedure is used

Citation: Journal of Atmospheric and Oceanic Technology 19, 12; 10.1175/1520-0426(2002)019<1968:IPCFAR>2.0.CO;2

Table 1. 

Calibration parameters used to calculate the distortion matrices D and

Table 1. 

1

Although one does not always need to compute all the elements of cs, in principle all the elements of the measured covariance vector cm should be used for computing any element of cs. However, in some cases the calibration procedure can be significantly simplified. For copolar measurements a sufficiently accurate estimate of any copolar-related element of cs can be estimated only by using measured copolar covariances and average copolar powers in cm. An accurate estimate of any cross-polar-like element of cs, on the other hand, will require measurements of all nine elements of cm. For more detailed analysis of errors that one would introduce by using only a few elements of cm, consult the magnitudes of the elements of the matrix −1.

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