## Abstract

A polarimetric weather radar with alternate transmission of slant linear +45° and −45° polarization and simultaneous reception of both linear vertical and linear horizontal polarization is considered. The equations of the radar observables for a model medium containing nonspherical hydrometeors are presented. Assuming the hydrometeors to be axially symmetric with a canting angle distribution symmetric about the mean canting angle, a set of equations for separation of propagation and backscattering effects is developed. The mean apparent canting angle, the degree of common orientation of the hydrometeors, and the differential phase shift are obtained. Using empirical relationships, the mean and differential attenuations are estimated by means of the differential phase shift. The intrinsic value of the reflectivity, the differential reflectivity, and the copolar correlation coefficient at zero lag time are then determined. Application of this to a model convective rain cell shows that the use of simultaneous transmission and reception of linear vertical and linear horizontal polarization at S, C, and X bands provides accurate estimates of the intrinsic scattering properties of precipitation. The analysis of the bias on the radar observables due to the assumption of a medium of equioriented hydrometeors shows that all the observables with the exception of reflectivity can be severely affected.

## 1. Introduction

It is now established that the use of polarization diversity in the radar study of precipitation improves quantitative estimation of rainfall, especially in heavy rain (cf. English et al. 1991; Ryzhkov and Zrnic 1995), and provides useful information for particle identification (cf. Hendry and Antar 1984; Balakrishnan and Zrnic 1990; Torlaschi and Holt 1993). Polarimetric radar has also demonstrated great potential in solving a variety of problems in operational meteorology such as identification of ground clutter and storm updraft regions, three-body scattering phenomena, and overcoming attenuation effects on rainfall estimated by short-wavelength radars (cf. Holt et al. 1994; Joe et al. 1995; Zrnic 1996).

Polarization diversity includes using any pair of orthogonal polarizations. In practice the only polarization schemes attempted are circular (McCormick and Hendry 1975), linear vertical/horizontal (V/H; Seliga and Bringi 1976), and slant linear (±45°) polarization (cf. Chandrasekar et al. 1994). One radar that has the capability of using all three schemes is the C-band Deutsches Zentrum für Luft und Raumfahrt (DLR) radar at Oberpfaffenhofen, Germany (Schroth et al. 1989). However, the polarization method implemented more frequently has been dual-linear H/V polarization. In this case H and V polarization is alternately transmitted and the copolar power is measured. The total differential phase and the copolar correlation coefficient are then estimated from Doppler processing of staggered measurements.

There is now work undertaken at the National Oceanic and Atmospheric Administration (NOAA) to provide polarimetric capability for the WSR-88D, and a novel polarimetric scheme employing simultaneous transmission of horizontally and vertically polarized waves is being implemented on the research facility at National Severe Storm Laboratory (NSSL; Zahrai and Zrnic 1997; Doviak et al. 1998). This polarization method corresponds to the use of a slant linear polarization on transmit and simultaneous reception of both H and V polarization components as proposed by Sachidananda and Zrnic (1985). According to Doviak et al. (1998), the main motivation for simultaneous transmission of the H and V polarization is to eliminate the high-power microwave switch, which has been a critical component in dual-linear polarization research radars.

When the precipitation induces propagation effects in polarimetric radar measurements, the issue is to estimate the intrinsic value of the radar observables by means of radar measurements. The number of the precipitation properties that are linked in the propagation and backscattering processes exceeds the number of radar measurements. Therefore, to estimate the value of the intrinsic radar observables, additional assumptions have to be made. At S band, for example, propagation effects in rain are principally due to differential phase. The use of dual-linear polarization to measure copolar return power when linear H/V polarization is transmitted has the advantage that there are no phase effects and only little mean and differential attenuation on horizontal and differential reflectivity. However, in other types of precipitation and at higher frequencies propagation effects are no longer negligible and must be estimated for quantitative interpretation of the radar observables.

In this paper we report on theoretical work done at the Cooperative Centre for Research in Mesometeorology to asses the technology of polarization diversity radars in view of polarimetric upgrades for the S- and X-band radars at the J. S. Marshall Radar Observatory of McGill University (Gingras 1997; Gingras et al. 1997). As in Doviak et al. (1998), we examine hybrid linear dual-polarization diversity radar transmitting slant linear polarization and receiving H and V polarization. Different from the NOAA/NSSL’s WSR-88D research radar, we consider alternate transmission capability and simultaneous reception of both H and V polarization. The system includes therefore a high power microwave switch on transmission and two receivers.

In section 2 we introduce the radar observables, which are commonly used to characterize the scattering ensemble, and we summarize the theory to calculate the radar echo signal covariances and cross covariances in the case of hybrid linear dual-polarization. In section 3 we present a set of equations for separation of the propagation and the backscattering effects. A model convective rain cell is used in section 4 to assess and illustrate the method for separation of propagation and backscattering effects at S, C, and X bands. The assumption of a medium of equioriented hydrometeors that is frequently done in the literature is discussed in section 5.

## 2. Radar observables and propagation parameters

Let us assume that backscattering from a single body is represented by the backscattering matrix **S** referred to linear V/H base vectors

The intrinsic radar observables from an ensemble of particles filling a radar resolution volume which can be obtained and that are commonly used to characterize the properties of the scattering ensemble are (cf. Doviak and Zrnic 1993): the radar reflectivity at linear vertical and horizontal polarization, *Z*_{V} and *Z*_{H},

where *λ* is the radar wavelength, *K*_{w} is the refraction factor for water [i.e., *K*_{w} = (*n*^{2} − 1)/(*n*^{2} + 2), where *n* is the complex index of refraction], and the angle brackets indicate expected values; the differential reflectivity *Z*_{DR} (Seliga and Bringi 1976)

and the copolar correlation coefficient at zero lag time, *ρ*_{HV},

where the asterisk denotes the complex conjugate. Even though the linear depolarization ratio (LDR) is frequently used in polarization radar observations it is disregarded in this work because the polarization scheme considered here does not allow its determination.

The radar echo signal from an ensemble of particles filling a radar resolution volume can be represented by the coherency matrix (cf. Bringi and Hendry 1990) and the covariance matrix of the received signals. For transmission of slant linear +45° and −45° polarization and reception of V and H polarization, the coherency matrices **J**_{+} and **J**_{−} are, respectively,

where, for example, V_{+} denotes the complex value of the signal amplitude for transmission of slant linear +45° polarization and reception of linear vertical polarization. Following Sachidananda and Zrnic (1989), we define the covariance matrix **R** of the received signals at lag time *T*_{s} as

where the matrix **R**^{T}_{b} is the transpose of **R**_{b}, *ψ*_{D} is the Doppler phase due to the radial velocity of the scatterers,

When linear V/H polarization is used as a basis, according to Torlaschi and Holt (1998) the radar equation for a single particle can be represented by

where *C* is a constant dependent on radar parameters, *r* is the distance from the radar, and the matrix **M** relates the transmitter output signal (*T*_{V}, *T*_{H}) to the received signal (V, H). For a medium where the hydrometeors are axially symmetric and their distribution of canting angle is uniform along the propagation path, the elements of the transmission matrix **M** are (Holt 1984)

where *d*_{V,H} are complex propagation parameters that represent the total one-way on-path phase shift and attenuation for the two linear and orthogonal polarizations, and the apparent canting angle *α* is the projection of the canting angle *θ* of the particle in the plane normal to the direction of propagation. Note that the cross polar components, *s*_{HV}, *s*_{VH}, of the backscattering matrix are zero for axially symmetric hydrometeors. Furthermore, the propagation parameter are defined as *d*_{V,H} = exp(−*jK*_{V,H}*r*), where *r* is the range, and *K*_{V,H} = *k*_{0} + *k*_{V,H} = *β*_{V,H} − *jγ*_{V,H} is the propagation constant, *β*_{V,H} is the specific phase lag, and *γ*_{V,H} is the specific attenuation of the wave traveling through the medium. From the definition of the two-way mean attenuation, *A* = 2(*γ*_{V} + *γ*_{H})*r,* the two-way differential attenuation, Δ*a* = 2(*γ*_{H} − *γ*_{V})*r,* and the two-way differential propagation phase shift, *ϕ*_{DP} = 2(*β*_{H} − *β*_{V})*r,* it has been shown by Torlaschi and Holt (1993) that

Note that the mean and differential attenuation relates to the power of the signal while differential phase to the signal amplitude.

The definition of slant linear polarization requires the specification of a base direction, namely, the direction with respect to which the angle of the slant linear polarization is measured. In this paper the unit vector along linear vertical polarization is chosen for base direction.

For unit power delivered to the antenna, the transmitter output components at slant linear +45° and −45° polarization are 1/2 (1, 1) and 1/2 (1, −1), respectively, and from (7) and (8) the components of the received signals are

The appendix gives the expression of the radar echo signal covariances and cross covariances obtained by substituting (9) and (10) in (5) and (6). For the estimate of the Doppler spectral moments, see Sachidananda and Zrnic (1989).

## 3. Separation of propagation and backscattering effects

The reduction of the radar echo signal covariances and cross covariances (A1a)–(A1f) to useful forms requires the consideration of specific precipitation models. We consider next a model where the canting angle distribution *T*(*α* − *α*) is symmetric about the mean apparent canting angle *α* throughout the region of precipitation. According to McCormick and Hendry (1975) and Hendry et al. (1987), we define two parameters representing the degree of common orientation of hydrometeors filling the backscattering volume

For model calculations of circular polarization quantities, Torlaschi et al. (1984) have set *α* = 0°, and *ρ*_{2} = 0.98 in rain and *ρ*_{2} = 0.35 in hail. In Table 1 values of *α*, *ρ*_{2} and *ρ*_{4} for various precipitation types are summarized. The table includes the results reported by Hendry et al. (1987) based on radar observations collected with the dual-channel X-band radar of the National Research Council of Canada at Ottawa, Ontario (cf. Bringi and Hendry 1990). Torlaschi et al. (1984) and Hendry et al. (1987) give analytical expressions of *ρ*_{2} and *ρ*_{4} for Gaussian-like probability distribution of canting angles with standard deviation *σ*_{α} and with limits set at −*π*/2 and *π*/2. The relationship between *ρ*_{2} and *ρ*_{4} for Gaussian-like distributions is shown in Fig. 1, and a polynomial fit between this variables is given by

Hendry et al. (1987) show also that there is little difference in the ratio *ρ*_{4}/*ρ*_{2} between Gaussian-like, rectangular, and triangular distributions of canting angles.

When switched slant linear transmission is used, from (A1) and (11) we define the following quantities:

where Φ = *ϕ*_{DP} + *δ* is the two-way total differential phase shift, and *δ* = arg〈*s*_{VV}*s*^{*}_{HH}〉 is the backscattering differential phase shift. It follows from (13) that

In the right-hand side of (14c) and (14d) the superior signs are taken when *R*_{VV} + *R*_{HH} ≥ 0, and the inferior signs when *R*_{VV} + *R*_{HH} < 0. Given *ρ*_{4}, the value of *ρ*_{2} is determined from (12) or from Fig. 1. Note that *R*_{VV} + *R*_{HH} < 0 is typical for rain when propagation effects are not predominant. The mean apparent canting angle *α* can also be obtained from (13d). By substitution of (14) in (2)–(4), it follows that the intrinsic radar observables are estimated by

where *A*_{V,H} = *A* ∓ Δ*A*/2 is the total two-way attenuation at linear V, H polarization, respectively; Δ*A* = 8.686 Δ*a* is the two-way differential attenuation in decibel units; and in the right-hand side of (15a) and (15b) the superior signs are taken when *R*_{VV} + *R*_{HH} ≥ 0; and the inferior signs when *R*_{VV} + *R*_{HH} < 0.

## 4. Model calculations

We present in this section numerical simulations of radar measurements for transmission at S, C, and X bands of switched slant linear polarizations and reception of V and H polarizations. In the calculations of the radar echo signal covariances and cross covariances (A1), the scattering amplitudes are obtained using the transmission matrix approach (cf. Oguchi 1983) for drop size distributions of Marshall and Palmer (1948) with the relationship between size and shape from Pruppacher and Pitter (1971) and maximum diameter 6 mm. The distribution of canting angle is Gaussian-like about *θ* = 0° with standard deviation *σ*_{θ} as a function of the equivolume drop diameter *d*_{e} (Torlaschi and Pettigrew 1990; Torlaschi and Holt 1993),

In the simulations we assume a radar beam at 0° elevation, a temperature of 10°C. Further details on the calculation procedure can be found in Torlaschi and Holt (1993) and Gingras (1997).

According to Balakrishnan and Zrnic (1989) and McGuinness and Holt (1989), we consider that the two-way attenuation at linear horizontal polarization, *A*_{H}, and the two-way differential attenuation, Δ*A,* are proportional to the two-way differential propagation phase shift *ϕ*_{DP}. These relationships are obtained using least square fits to the values of *A*_{H}, Δ*A* and *ϕ*_{DP} from the numerical simulations themselves and are presented in Table 2. Note that only at S band for raindrops *δ* is negligible and Φ is a good estimate of *ϕ*_{DP}. At shorter wavelengths, *δ* is not necessarily negligible and must be estimated for quantitative interpretation of *ϕ*_{DP}. The algorithmic processing of data to obtain *ϕ*_{DP} remains a problem to be solved. There have been attempts (Tan et al. 1991; Scarchilli et al. 1993), but a method of general application has yet to be found. In this paper we take the value of Φ as the estimate of *ϕ*_{DP} at all the radar wavelengths.

To assess the set of equations for separation of propagation and backscattering effects, the rainfall rate is specified as a function of range and the radar observables are calculated at every range by the following procedure.

According to Marshall and Palmer (1948) the rainfall rate is used to obtain the drop size distribution.

The drop size distributions and the scattering amplitudes of each drop category are used to calculate the backward-scattering parameters of the ensemble and the intrinsic radar observables Z

_{V,H}, Z_{DR}, and*ρ*_{HV}. In these calculations Gaussian-like model distributions of canting angle are used.The range dependence of the forward-scattering parameters is accounted in the calculations of the propagation parameters.

The radar echo signal covariances and cross covariances are calculated from (A1).

The apparent radar observables

*Z̃*_{V±}∼ 〈|V_{±}|^{2}〉,*Z̃*_{H±}∼ 〈|H_{±}|^{2}〉,*Z̃*_{DR±}= 10 log(〈|H_{±}|^{2}〉/〈|V_{±}|^{2}〉), and*ρ̃*_{HV±}= 〈V^{*}_{±}H_{±}〉/(〈|V_{±}|^{2}〉〈|H_{±}|^{2}〉)^{0.5}are calculated.The quantities

*p, q, u, υ,*and*w*are calculated from (13).The value of

*α*and*ρ*_{4}are calculated from (14a) and (14b).From (12) the value of

*ρ*_{2}corresponding to*ρ*_{4}is determined.The total differential phase shift is calculated from (14f).

The estimated values of the intrinsic radar observables

*Ẑ*_{V,H},*Ẑ*_{DR}and*ρ̂*_{HV}are then obtained from (15) and from the estimates of*A*_{H}, and Δ*A*by means of Φ and the linear regressions reported in Table 2.

Figure 3 shows for the model convective cell in Figure 2, respectively, the values at S, C, and X bands of *ρ*_{2}, and *ρ*_{4}; the intrinsic, apparent, and estimated values of reflectivity, differential reflectivity; and copolar correlation coefficient at zero lag time; and the values of the two-way total differential phase shift and the two-way differential propagation phase shift. Minimum detectable reflectivities at 10 km are taken to be equal to −3 dB*Z* at S band, and −10 dB*Z* both at C and at X bands. At X band the radar echo is below the minimum detectable signal beyond about 27 km range. As expected, the estimates of *α* are practically nil for all ranges and are not shown. The reader is also referred to Gingras (1997) for a detailed analysis of similar numerical simulations when a population of equioriented hydrometeors is assumed.

It appears from Figs. 3d–l that at C and X bands the propagation effects on reflectivity (*Z*_{H} − *Z̃*_{H}), differential reflectivity (*Z*_{DR} − *Z̃*_{DR}) and the copolar correlation coefficient at zero lag time (*ρ*_{HV} − *ρ̃*_{HV}) are extreme beyond about 20 km, while at S band are at most moderate on the entire extent of the precipitation except for *ρ*_{HV} at the far end. From our calculations it is seen that the maximum bias is about 3, 28, and 78 dB on reflectivity; about 0.5, 10, and 16 dB on differential reflectivity; and 0.44, 0.3, 0.02 on the copolar correlation coefficient at S, C, and X bands, respectively. The values at X band are from about 27-km range. The observables *Ẑ*_{H}, *Ẑ*_{DR}, and *ρ̂*_{HV} counterbalance for most of the large biases and provide good estimates of the intrinsic values. Part of the residual overestimation at middle ranges on reflectivity and differential reflectivity is due to the use of Φ instead of *ϕ*_{DP} in the calculations of the horizontal and differential attenuation. For example, in the middle of the convective cell the backscattering differential phase shift at X band is about 7°, and from Table 2 the corresponding overestimation of the horizontal and differential attenuation are 2 and 0.44 dB, respectively, which correspond to most of the difference between the intrinsic and the estimated values shown in the figure. At the far end of the precipitation *Ẑ*_{DR} and *ρ̂*_{HV} at C and X bands do not provide useful estimates of differential reflectivity and correlation. At X band this feature appears beyond 27 km. Figures 3a–c show that the orientation parameters *ρ*_{2} and *ρ*_{4} increase and decrease with range accordingly with the rainfall rate and (16). We note, however, that because of the simplifying assumptions leading to (14b), the estimated values of the orientation parameters do not reproduce the simmetry in the model convective cell. The use of a model of standard deviation of canting angle independent on drop size would have produced here slightly decreasing values of *ρ*_{2} and *ρ*_{4}. Figures 3m–o show that *ϕ*_{DP} on the far side of the detectable precipitation is about 137°, 291°, and 273° at S, C, and X bands, respectively, and that Φ from (14f) provides good estimates along most of the rain cell.

## 5. Radar observables and the degree of common orientation

When only one slant linear polarization state is used on transmit the parameters representing the radar echo signal are the powers at V and H polarization, the cross covariance between V and H polarization, and the Doppler spectral properties. The only precipitation model for the interpretation of the radar polarization measurements that can be adopted in this case is to assume a priori equioriented hydrometeors with *α* set to 0° throughout the region of precipitation. Subscript “0” in the following notation relates to it. From (A1a)–(A1f) we obtain

The radar observables are then estimated by substitution of (17) in (2)–(4). This polarization scheme is similar to the use of switched V and H transmission (cf. Doviak and Zrnic 1993). In both schemes the estimates of the intrinsic radar observables are obtained assuming *α* = 0°, but when slant linear is used, the total differential phase and the copolar correlation coefficient are estimated from simultaneous measurements, however only half of the power is available in each of the two receiver channels.

We examine next to which extent the assumption of equioriented hydrometeors may influence the estimate of the radar observables all other conditions being equal. From (15b) the apparent differential reflectivity, *Z̃*_{DR}, can be written as

Let us take the series expansion of (18) about |*q*|/*ρ*_{2}*p* = 0. For values of |*Z̃*_{DR}| ⩽ ∼5 dB all terms which are nonlinear in the perturbation *δ*(|*q*|/*ρ*_{2}*p*) can be neglected, thus

For a population of particles with same orientation throughout the region of precipitation, the degree of common orientation is unity and from (11)

From the substitution of (20) in (19), the apparent differential reflectivity for a population of equioriented hydrometeors can be approximated as

Table 3 gives values of the relative error of estimate, (*Z̃*_{DR0} − *Z̃*_{DR})/*Z̃*_{DR}, for various precipitation types. The values of *Z̃*_{DR} in the table are from Doviak and Zrnic (1993) and *ρ*_{2}’s are reproduced from Table 1. It appears that except for rain, differential reflectivity can be significantly underestimated when equioriented particles are assumed.

The difference between *Ẑ*_{H0} and *Ẑ*_{H} gives the absolute error of estimate when a population of equioriented hydrometeors is assumed. Proceeding from the logaritmic form of (15a) in a similar manner as for differential reflectivity and then substituting (19) in the result, we obtain that the absolute error of estimate of horizontal reflectivity is

Values of the absolute error of estimate, Δ*Ẑ*_{H}, for various precipitation types are summarized in Table 3. The assumption of equioriented particles is a cause of a slight underestimation of horizontal reflectivity, usually less than half of a decibel. For vertical polarization the expression of the absolute error of estimate is the same as in (23) except for the minus sign in front of the right hand term, and an overestimation of vertical reflectivity of less than half a decibel would be the usual consequence in this case.

From (14f) it is readily seen that

The absolute error of estimate, ΔΦ, is therefore a function of Φ and *ρ*_{2}. For 0 ⩽ *ρ*_{2} ⩽ 1, the maximum values of ΔΦ occurs when 45° ⩽ Φ ⩽ 90°. Table 3 gives the values of (ΔΦ)_{max} for various precipitation types. It appears that when the radar beam moves from a region of well oriented particles as in rain to a region of less oriented ones as in the melting layer, snow, or hail, the value of Φ_{0} may suddenly decrease. This suggests that part of the negatif radial gradients of Φ_{0} which are reported in the literature (cf. Watson and Torlaschi 1998) could be due to the variations of the degree of common orientation of the medium.

From (15c) and by the substitution of (14f), (24), (19), and (21) in the ratio between *ρ̂*_{HV0} and *ρ̂*_{HV}, we obtain

By means of series expansion of the second term on the right-hand side of (25) and neglecting all terms which are nonlinear in the perturbation of (*Z̃*_{DR}/8.686)^{2}, the ratio between *ρ̂*_{HV0} and *ρ̂*_{HV} is approximated by

The ratio *ρ̂*_{HV0}/*ρ̂*_{HV} has relative minimums for Φ = 90°. Table 3 gives for various precipitation types, values of the maximum relative errors of estimate, (*ρ̂*_{HV0} − *ρ̂*_{HV}/*ρ*_{HV})_{max}, that is, the values of the relative error of *ρ̂*_{HV} for Φ = 90°. Again except for rain, the assumption of en equioriented medium can lead to a severe underestimation of *ρ̂*_{HV}.

## 6. Conclusions

In this work we have considered the effects of propagation and backscattering for simultaneous sampling at V and H polarization of the received radar signals when slant linear +45° and −45° polarization is used on transmit. Given a precipitation model of axially symmetric hydrometeors with a canting angle distribution symmetric about the mean canting angle, we have shown how from radar measurements the mean apparent canting angle, the orientation factors, and the two-way total differential phase shift may be deduced and used to derive the values of reflectivity, differential reflectivity, and copolar correlation coefficient at zero lag time that relate to the intrinsic properties of the actual backscatter.

The mean apparent canting angle and the orientation factors are intrinsic properties of the precipitation region, do not depend on the radar frequency, and are estimated by specific combinations of the radar covariances and cross covariances. Hydrometeors constituting different precipitation media have different falling behaviors and a radar able to measure the mean apparent canting angle and the orientation factors as the one considered in this work will be able to identify the precipitation types.

Our calculations show that for the model convective rain cell in Fig. 2, S-band radars are moderately affected by propagation but severe underestimation of the radar observables is observed at C and X bands. Most of the biases are a consequence of the signal losses due to attenuation rather than depolarization and it is well-known that attenuation is much less severe at S band than at C and X bands. Attenuation and differential attenuation estimates by means of the two-way total differential phase shift together with combinations of the radar signal covariances and cross covariances allow most of the time to recover remarkably well the intrinsic radar observables at C and X bands in spite of the severity of the signal losses.

It is common practice to assume a medium of equioriented axially symmetric particles as closure assumption for the separation of propagation and backscattering effects. In this work we demonstrate that the orientation of the hydrometeors can lead to severe underestimation of differential reflectivity and copolar correlation coefficient at zero lag time in snow, the melting layer and hail. In particular, relative errors of estimate up to −65% and −40% for *Z*_{DR} and for *ρ*_{HV}, respectively, are not uncommon in hail. Furthermore, part of the large negative values of *ϕ*_{DP} radial gradients reported in the literature might be due to variations of the degree of common orientation of the medium when the radar beam moves from a region of well oriented particles to a region of less oriented ones and not to backscattering phase.

## Acknowledgments

This work was partially supported by the National Sciences and Engineering Research Council of Canada, and by the Atmospheric Environement Service Department of Environment Canada. Our particular thanks are due to I. Zawadzki from J. S. Marshall Radar Observatory, McGill University, Montreal, Canada, for much helpful interaction, and to D. S. Zrnic from NOAA, Environmental Research Laboratories, National Severe Storms Laboratory, Norman, Oklahoma, for providing the computer program for scattering amplitude calculations.

## REFERENCES

_{DR}measurement considerations for a fast scan capability radar. Radio Sci., 20, 907–922

### APPENDIX

#### Radar Echo Signal Covariances and Cross Covariances

From the substitution of (9) and (10) in (5) and (6), we obtain the following expressions for the covariances and cross covariances:

where *C*′ depends on radar parameters, *e*^{−A} = |*d*_{V}*d*_{H}|^{2} is the two-way mean attenuation, *e*^{Δa} = |*d*_{V}/*d*_{H}|^{2} is the two-way differential attenuation, Φ = *ϕ*_{DP} + *δ* is the two-way total differential phase shift, *ϕ*_{DP} = arg{*d*_{V}*d*^{*}_{H}/*d*^{*}_{V}*d*_{H}} is the two-way differential propagation phase shift, and *δ* = arg〈*s*_{VV}*s*^{*}_{HH}〉 is the backscattering differential phase shift. In the derivation of (A1) we have assumed that the particle canting angle and the particle backscattering phase shift are statistically independent.

## Footnotes

* Current affiliation: Environnement Canada, Centre Météorologique Canadien, Dorval, Quebec, Canada.

*Corresponding author address:* Enrico Torlaschi, Département des Sciences de la Terre et de l’Atmosphère, Université du Québec à Montréal, Case postale 8888, Succursale Centre-Ville, Montréal, PQ H3C 3P8, Canada.