Radar polarimetric measurements are made on an orthogonal measurement basis defined by feeders of the antenna system. In the hybrid mode the measurement basis is typically Cartesian and formed by the horizontal (h) and vertical (v) components. Further this basis is denoted as the h–v basis. Dual-polarimetric cloud radars have two receivers dedicated to the orthogonal polarimetric components of the received signal. For each transmitted pulse the receivers provide range profiles of in-phase Ih,v and quadrature Qh,v components, where indices h and v denote the polarization state. Note that this study does not cover the radar signal processing to get the Ih,v and Qh,v profiles. This information can be found in a radar handbook (e.g., , chap. 6). Using Nfft profiles of Ih+iQh and Iv+iQv, where i is the imaginary unit, the radar calculates complex Doppler spectra in the horizontal and vertical channel, respectively, applying the fast Fourier transformation (FFT) along the time dimension. The complex Doppler spectra are represented by complex amplitudes S˙ for each spectral component and each range bin. In the following, S˙h and S˙v denote the measured complex amplitudes of the analyzed spectral component in the horizontal and vertical channels, respectively (the dot hereafter denotes a complex quantity).

Different range bins as well as different spectral components are often considered to be statistically independent because the corresponding complex amplitudes result from non-coherent scattering of numerous independently moving particles. Some correlation, however, can be expected due to sampling effects and the FFT spectral leakages e.g., Sect. 5.3 in. For instance, the power scattered from particles located close to the end of a range bin is distributed between this and the following range bins. These effects depend on filter properties and used FFT windows. It is challenging to give a general analytical solution taking these effects into account. Therefore, these effects are out of the scope of this study. For the sake of simplicity the following analysis is shown only for a single range bin and a single spectral component. Since movements of particles in neighboring range and spectral bins are not related, statistical properties of an individual bin considered in the following are not affected by sampling effects and spectral leakages. The neglection of the dependence of the neighboring bins (due to sampling effects and spectral leakages) leads to an underestimation of the information entropy when a complete spectrum and/or spectral profile is analyzed. This worst case assumption, however, allows for a relatively easy and universal characterization of measurement errors. Future studies may improve the error characterization by considering the sampling and leakage effects.

Unlike precipitation radars which perform rapid azimuth scans, cloud radars are typically pointed to a certain direction or make slow scans to get non-broadened Doppler spectra. showed (Eq. 5.2 therein) that the coherency between the adjacent samples depends on the wavelength and the sample repetition period. Cloud radars typically have the pulse repetition frequency on the order of 10 kHz and Nftt in the range of 128 to 1024. This results in getting a single spectrum every 0.01–0.1 s. For such sampling properties of cloud radars any significant coherency between adjacent samples of a spectral line requires the spectral broadening not exceeding at most a few centimeters per second. The turbulent spectral broadening, however, exceeds a few centimeters per second even in stratiform non-precipitating clouds . Therefore, consecutive samples of complex amplitudes for a spectral line can be considered to be independent.

Introduce a measurement column vector 1m^=[R^h,J^h,R^v,J^v]T with R^ and J^ being real and imaginary parts of a complex amplitude S˙, indices h and v denoting the polarization state, and T being the transposition sign; the circumflex is used hereafter to emphasize measured quantities. The probability density function (PDF) of m^, given the true covariance matrix Σm of m^, can be written as follows: 2fmm^|Σm=(2π)-2det(Σm)-12e-12m^TΣmm^.

Note that throughout the study a PDF is a function of measured quantities (e.g., m^ in Eq. ) with fixed parameters (e.g., Σm in Eq. ). The same PDF is called a likelihood function if the measured quantities are fixed, and the PDF is viewed as a function of parameters.

showed that for meteorological targets I and Q components are jointly normal with zero mean, zero correlation, and equal standard deviation. The authors explain that these properties are due to scattering from a large number of particles moving in an unpredictable way in a scattering volume. Since Nfft is much smaller than the number of particles in a resolution volume, the properties are also valid for relations between R^h and J^h and between R^v and J^v.

The measured complex amplitudes S˙h and S˙v, however, can be correlated. Taking these properties into account, the true covariance matrix Σm is defined in the following way (Eq. 5.178 in ): 3Σm=σh20qσhσvsσhσv0σh2-sσhσvqσhσvqσhσv-sσhσvσv20sσhσvqσhσv0σv2,, where σh is the standard deviation of R^h and J^h, σv is the standard deviation of R^v and J^v, q is the correlation between R^h and R^v, and s is the correlation between R^h and J^v.

Since for meteorological targets R^h is not correlated with J^h, and R^v is not correlated with J^v, the absolute phases of S˙h and S˙v are uniformly distributed from 0 to 2π and thus uninformative. Therefore, the polarimetric observations in the hybrid mode can be represented by a 2×2 covariance matrix B (Eq. 4.130 in ) instead of the true covariance matrix Σm: 4B=eeT‾=BhhB˙hvB˙hv∗Bvv, where 5e=(S˙h,S˙v)T; the overline indicates the expected value; Bhh and Bvv represent total powers of the horizontal and vertical components of the received signal, respectively; B˙hv is the covariance between the horizontal and vertical components of the received signal; and ∗ is the complex conjugation sign. Note that in general Bhh, Bvv, and real and imaginary parts of B˙hv can be calibrated in any quantity that is proportional to the power (watts) received by the radar, e.g., classical radar reflectivity (mm6 m-3) or even arbitrary units . Recall that in this study the covariance matrix B corresponds to a single spectral component. Such spectral representation of vector signals was introduced by .

The elements of B are related to the statistics of the complex amplitudes S˙h and S˙v as follows: 6Bhh=var(R^h)+var(J^h)=2σh2,7Bvv=var(R^v)+var(J^v)=2σv2,8B˙hv=Rhv+iJhv=(q+js)σhσv, where Rhv and Jhv are real and imaginary parts of B˙hv.

In the precipitation radar community, dual-polarized measurements are rarely represented by B. Instead a set of polarimetric variables are used. Therefore, the same polarimetric variables (but spectrally resolved) are introduced in this study. Introduce a vector 9c=(Bhh,ZDR,ρHV,ΦDP)T, where ZDR is the differential reflectivity, ρHV is the correlation coefficient, and ΦDP is the differential phase. In this study ZDR, ρHV, and ΦDP are defined for each spectral line using elements of corresponding B: 10ZDR=BhhBvv,11ρHV=Rhv2+Jhv2BhhBvv,12ΦDP=atan-JhvRhv.

Note that elements of the matrix B are in general affected by noise. The noise in both polarimetric channels is not known exactly. Typically, it is estimated from spectra using, for example, the algorithm from . A subtraction of noise levels from corresponding diagonal terms of the covariance matrix B to get an estimate of signal-only powers leads to occasions when the covariance matrix is no longer positively semi-definite. In this case, ρHV calculated from the noise-corrected covariance matrix can exceed 1, which is beyond the range of valid values. In order to avoid this problem, we characterize radar measurements without noise subtraction. A further advantage of this approach is that spectral lines containing noise only can also be correctly characterized.

As previously mentioned, S˙h and S˙v are, in general, correlated. There is, however, always a basis on which the projections of S˙h and S˙v become completely uncorrelated. This basis is further denoted as the c–x (co-polar and cross-polar) basis. The conversion of the vector e on the h–v basis to the vector eD on c–x basis is made using the unitary operator Q: 19eD=S˙cS˙x=Qe. The calculation of the matrix Q is given in Appendix . Real and imaginary parts of S˙c are jointly distributed normally with the zero mean, zero correlation, and standard deviation σc. Real and imaginary parts of S˙x are also jointly distributed normally with zero mean and zero correlation but have, in general, a different standard deviation σx.

A transformation from the basis h–v to the basis c–x also changes the covariance matrix of the measurements. The covariance matrix D of measurements on the c–x basis is diagonal and can be found as follows: 20D=Dcc00Dxx=Q†BQ.

In Eq. () † is the Hermitian conjugate. Zero off-diagonal terms in D indicate that there is no correlation between the orthogonal components, i.e., S˙c and S˙x. Expanding Eq. (), the elements of the matrix D can be found as follows: 21Dcc=q112Bhh+q˙122Bvv-2q11R12Rhv+J12Jhv22Dxx=q˙122Bhh+q112Bvv+2q11R12Rhv+J12Jhv, where q˙nm represents elements of Q, with n and m being indices of row and column, respectively; 23q˙12=R12+iJ12.

Similar to relations between the powers and the standard deviations given in Eqs. () and (), σ1 and σ2 are related to Dcc and Dxx, respectively: 24Dcc=var(Rc)+var(Jc)=2σc225Dxx=var(Rx)+var(Jx)=2σx2.

The measured values D^cc, 26D^cx=R^cx+iJ^cx, and D^xx represent elements of the matrix D^: 27D^=Q†B^Q. Note that the operator Q here is the same as in Eq. () and not recalculated using B^.

In the previous step, measurements were represented on a new – c–x – basis on which the orthogonal components of the measurement vector are independent. On this basis the joint multivariate likelihood function can be represented as a product of likelihood functions with a single element as an argument. This allows for a relatively easy mathematical description of the likelihood function on the c–x basis.

By definition, the off-diagonal elements of the covariance matrix D are zeros (see Eq. ). This implies no correlation between S˙c and S˙x. In this case, the likelihood function fd(d^|b,Ns), where 28d^=D^cc,R^cx,J^cx,D^xxT, can be written as a multiplication of likelihood functions of individual components: 29fd(d^|b,Ns)=fD^cc|b,NsfR^cx|b,Ns×fJ^cx|b,NsfD^xx|b,Ns.

The derivation of the formulas for the calculation of the likelihood functions is tedious and provided in full in Appendix  for the interested reader. The likelihoods of the individual components can be computed as follows: 30fD^cc|b,Ns=Nsσc2χ2Ns2Nsσc2D^cc,31fD^xx|b,Ns=Nsσx2χ2Ns2Nsσx2D^xx, 32fR^cx|b,Ns=(2Ns)a|R^cx|-bπ22Ns(σcσx)aΓ(Ns)Kb×|2NsR^cx|σcσx,33fJ^cx|b,Ns=(2Ns)a|J^cx|-bπ22Ns(σcσx)aΓ(Ns)Kb×|2NsJ^cx|σcσx, where χk2 is the chi-squared distribution with k degrees of freedom, 34a=(2Ns+1)/2,35b=(1-2Ns)/2, Γ is the gamma function, and Kμ is the Bessel function of the second kind of order μ. Recall that σc and σx in Eqs. ()–() are derived from the elements of b using Eqs. ()–() and Eqs. () and (). Derivation and Monte Carlo evaluation of Eqs. ()–() are given in Appendix . Appendix  shows how to handle Eqs. () and () when R^cx and J^cx are close to 0.

In the previous step, the likelihood function of measurements represented on the c–x basis was derived. In this subsection we perform a transformation back from the c–x basis to the original h–v basis that allows for the comparison of radar measurements in a common orthogonal reference frame.

Applying the rule of changing variables in a multivariate PDF e.g., Theorem 7.4, fb(b^|b,Ns) can be found from Eq. () as follows: 36fb(b^|b,Ns)=|Jbd|fd(d^|b,Ns).

As shown in Appendix , the determinant of the Jacobian Jbd of the transformation from b^ to d^ is equal to 1.

As already mentioned, the polarimetric measurements are commonly described by means of a set of quantities (ZDR, ρHV, ΦDP) that are non-linear functions of the elements of B. It is therefore interesting to derive the likelihood function of those quantities.

Likelihood fc(c^|b,Ns) of a vector 37c^=(B^hh,Z^DR,ρ^HV,Φ^DP) can be found by multiplying fb(b^|b,Ns) by |Jcb|, with 38Jcb=-Bhh3ZDR-3ρHV being the Jacobian of the transformation from c^ to b^ (see Appendix ): 39fc(c^|b,Ns)=Bhh3ZDR-3ρHVfb(b^|b,Ns).

The final results of this section – Eqs. () and () – can be used for the maximum likelihood optimization and Bayesian inference methods. Ready-to-use MATLAB implementations of these equations are provided in the Supplement.

In this section we start from the representation of measurements on the c–x basis because the measurement errors are independent in this case. Recall that Eq. () relates the covariance matrix D^ and B^. This equation thus can be used to find relations between elements of the vector b^ on the original h–v basis and elements of the vector d^ on the c–x basis.

The covariance matrix B^ estimated from measurements is related to the matrix D^ as follows: 40B^=QD^Q†.

After expanding Eq. () it can be seen that the elements of the vector b^ can be found as linear combinations of the elements of the vector d^: 41B^hh=q112D^cc+q˙122D^xx+2q11R12R^cx+J12J^cx,42R^hv=q11R12D^xx-D^cc+q112-R122+J122R^cx-2R12J12J^cx

43J^hv=q11J12D^xx-D^cc+q112+R122-J122J^cx-2R12J12R^cx44D^vv=q˙122D^cc+q112D^xx-2q11R12R^cx+J12J^cx. Or they can be found in matrix form: 45b^=q1122q11R122q11J12q˙122-q11R12q112-R122+J122-2R12J12q11R12-q11J12-2R12J12q112+R122-J122q11J12q˙122-2q11R12-2q11J12q112d^=Md^.

In this case, as shown in Wilks (chap. 10.4.3), the error covariance matrix Σb of b^ can be calculated from the error covariance matrix Σd of d^: 46Σb=MΣdMT, where 47Σd=4σc4/Ns0000σc2σx2/Ns0000σc2σx2/Ns00004σx4/Ns.

Recall that the off-diagonal terms of Σd are set to 0 taking into account that the elements of d^ are not correlated. The derivation of diagonal terms – variances of elements of d^ – is given in Appendix .

The main result of this subsection – Eq. () – was implemented as a ready-to-use MATLAB function that is available in the Supplement.

In the next subsection we also consider the error covariance matrix of the vector c^ – the conventional representation of polarimetric measurements. Note however that, as is shown in Sect. , the approximation of the error covariance matrix of c^ has issues which may limit its applicability.

As is shown in Sect. , the error covariance matrix Σb can be used to characterize uncertainties in spectral radar observations. By analogy to what is done in Sect. , one might think about applying again the rules of linear transformation to obtain the error covariance of the vector c^. In this section we do that by means of a linearization of the formulas that define the components of c (Eqs. , , and ). It is further demonstrated in Sect.  that such representation of measurement uncertainties for c^ is deficient.

Recall that the calculation of c^ includes highly non-linear functions (see Sect. ). Therefore, the error covariance matrix Σc of the vector c^ is estimated using the first-order Taylor approximation: 48Σc=SΣbST, where S is the sensitivity matrix: 49S=∂Bhh∂Bhh∂Bhh∂Rhv∂Bhh∂Jhv∂Bhh∂Bvv∂ZDR∂Bhh∂ZDR∂Rhv∂ZDR∂Jhv∂ZDR∂Bvv∂ρHV∂Bhh∂ρHV∂Rhv∂ρHV∂Jhv∂ρHV∂Bvv∂ΦDP∂Bhh∂ΦDP∂Rhv∂ΦDP∂Jhv∂ΦDP∂Bvv.

Note that the utilization of the first-order Taylor approximation for variances of polarimetric variables was proposed in the classical book of . Equation (D1) in Appendix D shows the complete matrix S in terms of Bhh, Bvv, B˙hv, Rhv, and Jhv.

A ready-to-use MATLAB implementation of Eq. () is provided in the Supplement. As is shown in the next section, the error covariance matrix Σc does not always reflects the true statistical properties of polarimetric observations. Therefore, this approximation is provided only for demonstration purposes, and it is not recommended.

All I/Q measurements within a chirp sequence in every polarimetric channel are split into 224 continuous blocks. Each block contains 32 I/Q pairs. The FFT with the Blackman weighting window is applied to each block to get complex Doppler spectra. Then the 224 blocks are split into 28 sub-blocks with 8 spectra in each sub-block. Within each sub-block elements of the vector b^ are calculated according to Eqs. ()–() with Ns=8 for every spectral line. For each b^ the vector c^ is obtained. Note that for this, Eqs. ()–() were applied to elements of b^ instead of b. Using vectors b^ and c^ within a sequence, the error covariance matrices Σ^b and Σ^c are calculated numerically. The circumflex here indicates that the error covariance matrices are estimated from measurements.

The calculation of the likelihood functions using Eqs. () and () requires b. The approximation of covariance matrices using Eqs. () and () requires the matrix B. In order to estimate b and B, elements of the vector b^ are averaged over 28 sub-blocks available within a single chirp sequence. These averaged values are assumed to be elements of the vector b from which the matrix B is obtained. Using B and Ns=8, Σb and Σc are calculated for each chirp sequence as shown in Fig. .

The random error analysis provided in this study is only applicable to volume-distributed scattering and noise. As discussed in Sect. , in this case R^h is not correlated with J^h, and R^v is not correlated with J^v. However, radar observations in general contain scattering from atmospheric plankton, ground clutter, and coherent receiver noise, which do not fulfill the assumption. In order to filter out spectral lines with correlated real and imaginary parts, a simple filtering rule was applied. It is known that for a signal with uncorrelated in-phase and quadrature components, its mean power and power standard deviation are related to each other Eq. 5.193 in. Figure  shows distributions of the mean power over the power standard deviation calculated in the horizontal and vertical polarization channels shown by blue and yellow lines, respectively. It can be seen that the mode of the distributions is close to the theoretical value of Ns=2.8. The distributions, however, have a considerable tail on the left side. These small values of the ratio are expected for correlated in-phase and quadrature components. Thus, a threshold in the ratio of the mean power over the standard deviation of power can be used to filter out unwanted spectral lines. In order to specify the threshold, the Monte Carlo approach was used. A total of 15.9×106 random complex values with normal distribution, zero mean, and a standard deviation of 1 were generated. The same processing as for measured I/Q data was applied to the generated complex values. The distribution of the ratio of the mean power over the power standard deviation for the generated data (denoted as expected distribution) is shown in Fig.  by the red line. The expected distribution has a much smaller tail on the left side relative to the ones of the measured distributions. The threshold of 2.3 used for filtering is chosen as the 5th percentile of the expected distribution. Vectors b^ and c^ are excluded from the analysis if for the corresponding spectral component within a chirp sequence the ratio of the mean power over the power standard deviation is below the threshold in at least one of the polarimetric channels. Around 18 % of the data are excluded.

Recall that b is estimated from measurements by averaging all available sub-blocks within a chirp sequence; b, however, can also be estimated by maximization of the likelihood functions given in Eqs. () and (). In this case, an optimization algorithm needs to be employed to find a set of elements of b corresponding to the global maximum in either Eq. () or Eq. (). This study uses a derivative-free optimization method available by default in MATLAB . Since the optimization method minimizes a function, the likelihood functions were not used directly. Instead, the following cost functions were used for the minimization:

50Cb=-∑l=128log⁡10fbb^|b,Ns,51Cc=-∑l=128log⁡10fcc^|b,Ns.

Here the index l runs over 28 sub-blocks within a chirp sequence. Equations () and () take into account that the consecutive b^ vectors are not correlated. In this case the total likelihood of 28 vector b^'s is a product of likelihood of each individual b^. In order to avoid an overflow of double numbers, the logarithm was used. In this case the logarithm of the product is replaced by the sum of logarithms. The logarithm is a monotonically increasing function, and therefore it does not change the position of the maximum of the likelihood function. Finally, the minus sign was introduced to have a smaller value of a cost function corresponding to a higher value of the likelihood. For the evaluation, 1000 chirp sequences were chosen randomly for the maximum likelihood estimation using fb(b^|b,Ns). In each chirp sequence a single spectral line was randomly chosen for the analysis. Thus, there are 28 vector b^'s available in each of the 1000 chirp sequences. For each sequence, the optimization algorithm requires an initial guess of b. In order to avoid local minima, five different initial guesses were used, which are a coefficient P multiplied by the first b^ in the analyzed chirp sequence. The values of P were 0.5, 0.75, 1, 1.25, and 1.5. The solution giving the lowest cost function out of the five outcomes was chosen as the result. Similarly the maximum likelihood estimation using fc(c^|b,Ns) was done using independently chosen 1000 chirp sequences. Figure  shows a comparison of elements of b estimated by averaging over 28 sub-blocks and those estimated by the maximum likelihood approach. All panels show a good agreement indicated by the close-to-unity slope of the linear regression. Both fb(b^|b,Ns) (results in the first row of Fig. ) and fc(c^|b,Ns) (results in the second row of Fig. ) show the same level of agreement and, therefore, can be used with no difference.

Diagonal elements of Σc – variances of B^hh, Z^DR, ρ^HV, and Φ^DP – were checked against those calculated using Eqs. (6.139a), (6.141), (6.144), and (6.143) in , respectively. Taking into account that samples for a spectral line are not correlated, approximations for variances of B^hh, Z^DR, ρ^HV, and Φ^DP based on the equations in are

52VARbhh=Bhh2Ns,53VARzdr=2ZDR2(1-ρHV2)Ns,54VARρ=(1-ρHV2)22NsρHV2,and55VARΦ=(1-ρHV2)2NsρHV2, respectively.

Figure  shows that VARbhh, VARzdr, and VARΦ match exactly Σc(1,1), Σc(2,2), and Σc(4,4), respectively. VARρ, however, agrees with Σc(3,3) only at values of ρHV>0.95. Below this value VARρ overestimates the variance of ρ^HV. At values of ρHV close to 0, VARρ has unrealistically high values, which result from ρHV in the denominator of Eq. ().

Figure d also shows unrealistic values with both approximations of the Φ^DP variance. Taking into account that Φ^DP can take values within the range of 0 to 2π rad, the variance of Φ^DP exceeding 103 rad2 is definitely erroneous. The high variance of Φ^DP corresponds to values of ρHV<0.3. This effect results from the first-order Taylor approximation of Eq. (), which is a highly non-linear function.

A comparison of the error covariance matrices Σ^c with the calculated one Σc is shown in Fig. . Figure f, k, and p indicate considerable differences caused by the first-order Taylor approximation in variances of Z^DR, ρ^HV, and Φ^DP, respectively. The results also reveal that the first-order Taylor approximation cannot adequately represent most of the non-diagonal components of the error covariance matrix.

Figure  shows a comparison of elements of error covariance matrices Σ^b estimated from the radar measurements with those calculated using Eq. (). Estimated and calculated elements are in a good agreement. Linear regressions shown in the panels by red lines have slopes close to 1. Pearson correlations between estimated and calculated elements exceed 0.96. These results indicate an agreement of the theoretical calculation with measurements and thus confirm the correctness of Eq. (). As expected, Figs. a and a show equivalent results. This is because the co-polar signal Bhh is effectively the same in both measurement representations and highlights the relevance of the present study only for dual-polarimetric quantities.

It is thus concluded that any application of spectral polarimetric measurements which requires the estimate of the error covariance matrix (e.g., variational retrievals, data assimilation, and sensitivity analysis) should be performed in the space of observations b^ rather than c^.