Report on Polarisation Weather Radar by A. R. Holt & D. H. O. Bebbington
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2. Representation and Analysis of PolarizationIn weather radar, polarization techniques grew piecemeal, and usually involved limited sets of parameters. Even in the 1980’s, the level of complexity of the electronic systems needed to offer full polarimetric processing, and in particular the requirements for real-time data processing and storage were such that they could not be realistically contemplated in an operational context. Thus, those polarimetric systems that were built involved considerable compromises in terms of what could be measured. Even the DLR polarimetric weather radar (ref. 22) does not measure the full correlation matrices in real time, and was restricted in data acquisition volume for full-matrix coherent measurement. In more typical weather radar systems, reflectivity (Z) and differential reflectivity (ZDR) were the norm, (requiring only copolar measurements, and therefore just one receive channel), while in some systems the linear depolarization ratio (LDR) was also measurable through the provision of two independent receive channels. Consequently, the analysis of weather radar data has typically been expressed in terms of the measurable parameters that these systems provided, with the addition of supplementary parameters in the case of systems with more elaborate hardware. In turn, the debate about what polarimetric capabilities are worth providing has also been conducted around these same parameters. 2.1 Coherency Matrices and Stokes Vectors Given a deterministic or partially coherent random vector electric field E with complex analytic signal representation and time signature  (1)
where the overbar denotes complex conjugation, and angle brackets denote averaging (ensemble, but equivalent to time averaging, assuming ergodicity) which is required for a stochastic field. The four-component Stokes vector (ref. 23)  is then defined by
 (2)
when the polarization bases are circular. Since the coherency matrix is Hermitian the Stokes vector is real. The term  (3)
This can conveniently be expressed as a quadratic form,  (4)
and if the components of the Stokes vector be interpreted as homogeneous projective coordinates, equation (4) describes the surface of a unit sphere in a standard orthogonal coordinate system. In the optics and electromagnetic literature this is known as the Poincaré sphere. On this sphere, orthogonal (basis) pairs are situated antipodally. In the conventional representation, left hand circular polarization is located at the north pole, right hand at the south, while linear polarizations occupy the equator: The axis of H/V polarizations lies at right angles to that of the 45˚ slant polarizations, and a spatial rotation corresponds to a rotation of the Poincare sphere by a double angle in general. This representation was used by McCormick and Hendry (ref .2) to depict optimal or near null polarizations, and by Bebbington et al (ref. 5) to analyze and correct for propagation effects due to differential phase at S-band, but has not been extensively used in the field of Radar Meteorology since. More recently, Krehbiel (ref. 24) has used it to describe depolarizing and alignment effects in thunderstorms. Fields due to independent scattering add linearly, and if the positions of scatterers are uncorrelated, as is believed to be the case in precipitation, the form of the coherency matrix can be broken down into contributions,  (5)
that involve self-correlations and intercorrelations of fields scattered by pairs of scatterers indexed by m and n with different scattering phases in general when the indices are distinct. Although the pair-wise terms arising from products between distinct particles are in practice far more numerous than those corresponding to a particle taken twice, their expectation is zero, and the averaged coherency matrix measured is evaluated as the sum of the terms that are due to single scatterers. Thus the coherency matrices and Stokes vectors are additive over the scatterers. Because of the well-known Cauchy inequality,  (6)
the determinant of a complex Hermitian coherency matrix is positive semi-definite. This condition translates into the equivalent property for the Stokes vectors,  (7)
so that in the geometric interpretation they lie on or within the sphere. A convenient way to envisage this is to consider the net Stokes vector to be the summation of a random walk of magnitude-scaled vectors from each scatterer. If the vectors are not perfectly aligned, the length of the resultant will be less than the sum of their lengths. Although the signal returned by a volume of preciptitation scatterers is intrinsically very random, its correlation properties are actually usually rather high, (for pure rain, realistic models show that the degree of polarization is usually > 98%) (ref. 25). In other words, the directions of the Stokes received vectors for any fixed transmission polarization are highly aligned. This property allows the polarimetric properties of a volume of hydrometeors to be largely understood as if describing a single scatterer. In the case of a true single scatterer, there are two co-polar null polarizations, that is, polarizations for which the co-polar power is zero. All the scattering properties of such a scatterer can be understood in relation to the geometrical disposition of these nulls; these and the bisector of their radii on the sphere are commonly referred to in polarimetric literature as the Huynen fork (refs. 26, 27). While this concept has not gained wide exposure within the Meteorological Radar community, it is capable of providing very useful insight into the phenomenology of polarization response in scattering from hydrometeors: firstly in terms of the effect of increasing drop size, and secondly in terms of propagation effects. In the case of a perfect sphere, the copolar response is null for circular polarizations. As drops increase in size their horizontal co-polar response increases with respect to the vertical. This is an effect equivalent to observing a spherical drop through a medium that attenuates more in the vertical. As the differential effect increases, the loci of the copolar null polarizations migrate towards the Stokes vector representing vertical polarization. That is, the Huynen fork’s prongs close up towards the vertical polarization point on the equator of the sphere. When observing a scatterer through a medium with differential attenuation (with aligned axes) it is not possible to distinguish between the effects of the scatterer differential reflectivity and those of differential attenuation. On the other hand, differential phase propagation results in a rotation of the Huynen fork about the axis corresponding to the H/V basis. While this is indistinguishable from backscatter differential phase, the fact that backscatter phase variations are generally neither large nor progressive explains why differential phase techniques for correcting for propagation effects have proven to be effective. Physical canting of scatterers results, as has been remarked, in rotation of the figure of the fork about the polar axis of the sphere, whci is distinct from the motions associated with the common propagation effects. In principle, it is possible to interpret many of the physical effects in the observation of hydrometerors if the full scattering matrix is measured. To this end, it is important to note that it would not matter in principle what polarization basis is chosen for the measurement. Unitary basis transformations correspond to ordinary spatial rotations of the Poincaré sphere, and the null-polarization map can be derived directly from the scattering matrix in an invariant way. The conventional usage in the Radar Meteorology community of ‘linear’ or ‘circular’ polarization parameters does not accurately reflect the information derivable except insofar as partial polarization information is measured. The key differences arise from whether both basis polarizations are transmitted, and/or both received. In a coherent system, this is the only major difference. The possibilities of gathering almost all the relevant information from partial measurements (and therefore more economically) rest with symmetry properties or assumptions. Thus, on the assumption that the hydrometeors are well aligned with H and V, a circular polarization measurement using one transmit polarization is adequate, as transmission of the opposite hand would yield the same information, except for an orientation dependent phase factor. In the case of incoherent systems, there is also a question of measurement of the cross-correlations between channels. Discussion of this point would lead to analysis not only of geometric parameters but of scatterer correlations, reflecting the larger number of degrees of freedom in assigning the Mueller matrix (van de Hulst) which related the returned Stokes vector to the transmitted one. The DLR radar appears to be the only weather radar to have been equipped to make full Mueller matrix measurements. However, it may be noted that for any coherent radar that measures the full scattering matrix, it is possible through signal processing – averaging of cross-products – to construct the empirical Mueller matrix. From this, any of the weather radar parameters associated with particular basis polarizations can be straightforwardly derived. 2.2 System Polarization Basis Currently, linear (H/V) polarization systems are far more prevalent than circular (L/R). There are also some experimental systems such as at CSU (ref. 28), which have some degree of flexibility in the polarization basis. Recently there has also been interest in systems in which the transmitter transmits states of polarization that are not aligned with the simultaneous receive channels. Some confused nomenclature has also arrived in the literature, such as the concept of ‘simultaneous H and V transmission’, which is technically erroneous as well as confusing. It is therefore important to set out a clear picture of how the choice of system polarization capability influences what can be measured, and perhaps most importantly to establish what in fact most matters. In this one whould consider both ideal situations as well as practical considerations, as technology is rapidly permitting closer approximations to the ideal. 2.2 Coherent measurement Modern radar systems are increasingly adopting coherent measurement. Early radars achieved their dynamic range through the use of logarithmic receivers, with phase information derived when necessary by the use of correlation of clipped or limited signals and the application of theory to the correlation properties of clipped gaussian noise (e.g. as in ref 29, 30). The performance – speed and accuracy – of modern analogue to digital converters (ADC) is such that with careful system design the required linear dynamic range for weather radar returns can be obtained in direct coherent sampling of the returned signal. This means that doppler processing and correlations in polarimetric radars can be performed through digital signal processing. In the case of a polarimetric radar, provided the transmit polarization state is switched at a rate that is significantly faster than the inverse decorrelation time of the hydrometeor volume (typically a few ms at C-band), coherent measurement allows the complete scattering matrix to be obtained. In the limit that the scattering volume is frozen, it would not matter in principle which polarization basis were chosen, because the same information is available, and the scattering matrices are related by unitary transformations. For example,  (8)
illustrates the (LR) representation of an ideally aligned hydrometeor described by transforming the inner HV-basis matrix on the r.h.s; the circular cross-polar term is much larger than the copolar terms, representing a CDR value of -20 log10(0.2) = -14 dB , for a ZDR of 20 log10(1.5) = 3.5 dB. In practice, because hydrometeors are moving, the scattering matrix is estimated from non-simultaneous sampling. In this case, there are consequences as what polarization scheme is used. If the scatterer has a radial velocity of 10 m/s, the phase error when sampled 1 ms later is of the order of 60 degrees at C-band. If there is a uniform Doppler velocity that the system measures, the phase error may be corrected. However, the estimate of the mean Doppler may not be very robust, so there will be a considerable error in the differential phase of the co-polar components. Differential propagation phase is increasingly used in (differential) attenuation correction, but these methods require rather accurate knowledge of the scattering differential phase. In a radar using circular basis, the cross-polar elements of the scattering matrix would be largest. Because of the symmetry of the matrix, the difference in phase can be attributed to the Doppler phase alone (ref. 31), and will have a better signal to noise ratio than any cross polar terms in an H/V measurement. In other words, this scheme offers the same equivalent pulse repetition rate as an unswitched radar, and as no statistical contribution to differential (H/V) scattering phase when the H/V basis scattering matrix is reconstructed.
It is widely considered that circular polarization systems are at a considerable disadvantage with respect to linear (H/V) in the weak-signal or long range regime because the copolar signal is so much weaker than the cross polar for standard hydrometeor targets, and may therefore appear to be submerged in noise. However, in a linear signal processing system there is no information loss, and it should therefore be possible to derive the same parameters through appropriate signal processing. Also, in practice, if in the weak signal regime one is prepared to accept that a theoretical estimate of the the degree of polarization is likely to be much better than the noisy measurement, then the magnitude of the weak co-polar channel can be quite well estimated using the other measured elements of the coherency matrix. This carries the advantage that in this way, information about the differential reflectivity, ZDR, is obtained simultaneously without incurring an error due to decorrelation as in the conventional linear H/V alternation technique – in fact this approach resembles the ‘one-shot ZDR’ method of Zrnic (ref H34). Further practical considerations in relation to the relative merits of circular and linear bases revolve around the question of the receiver amplitude and phase linearities over the dynamic range. In reality, rf amplifiers are not completely linear. In practice their specification will be such that at the limits of their specified dynamic range, some level of slope gain compression, e.g. 1 dB, will have occurred. Gain should be considered as a complex scalar, and it will normally be the case that the phase characteristic will also change over the entire dynamic range. In practice, most of this should occur rather close to the point where amplitude compression becomes noticeable. For dual channel receivers, there is an additional requirement that the amplifiers be gain and phase matched with each other. In engineering terms this is still somewhat challenging, as microwave device characteristics are extremely dependent on variations in fabrication tolerance, and so can rarely be exactly matched. One of the engineering arguments in favour of the use of linear polarization is that the amplitudes in the two channels are nearly always within a few decibels of each other, so that the effects of differential phase and gain differences remain relatively constant across the dynamic range. In balancing the scientific and engineering arguments over the relative merits of the two basic polarization approaches, one should bear in mind that the electronic processing of the received data can be considered as relatively independent of the actual transmission scheme. For example, while a circularly polarized system may present two ports representing the decomposition of the returned signal into circularly polarized components, these can be combined through the use of passive hybrid junctions so that the antenna appears as a linear receiver. Equally, and perhaps more simply, the transmitter output could be fed through a similar hybrid to a linear antenna to effectively transmit circular polarizations, while receiving linear. In the end, one can record two linear channels without theoretical loss of information and process the recorded data in any way that is convenient. In addition, modern rf switching technology for low power levels (using, e.g. PIN diodes) can be done with low loss, and good levels of crosstalk allowing cross-switching of the channel hardware to eliminate biases between the channels.
Polarisation weather radars are capable of making measurements of several properties of the back-scattered field, such as components of the coherency matrix or elements of the Mueller matrix. The number of such measurements that are possible with a given system will depend on the complexity of the particular radar system, and the particular measurements available will depend on the polarisation basis being used. All measurements necessarily involve a time-average of the received field measurements corresponding to a number of transmit pulses. In order to relate time-averaged measurements to “instantaneous” scattering matrices, it is necessary for the radar observation time to be small compared to the coherency time of the medium (TD). The latter is a function of the radar wavelength and the standard deviation V of the (assumed Gaussian) probability distribution of the radial velocity of the precipitation particles as 
For a C-band radar ( = 5.5cm) with  , TD ~ 2.06 10-3s.
Linear polarised radar systems need to switch polarisation states alternately in order to make measurements. Modern dual-linear polarised radar systems can switch transmission at time intervals of a few milliseconds, and hence this is much smaller than the coherency time of the medium. The radar parameters that are commonly derived from weather radar measurements are ZDR = Differential Reflectivity LDR = Linear Depolarisation Ratio KDP = Differential Propagation Phase ρHV (= ρ10) = Linear copolar correlation coefficient CDR = Circular Depolarisation Ratio p = Degree of polarisation (for circularly polarised systems) ORTT = “apparent degree of orientation” measured by circularly polarised systems The radar parameters relate to integrals of scattering terms over drop size distribution elevation and orientation angles, and types of particle within the resolution volume. Moreover most are also dependent on the effects of propagation along the path from antenna to resolution volume and back. For raindrops, it is normally assumed that the drops fall with their axes close to the vertical. It has been shown to be a reasonable approximation to assume that their axes are vertical, except for those parameters (e.g. LDR) which attain special values when the axes are precisely vertical. Another assumption often made is that the radar beam is elevated at 0 elevation, and hence that the radar beam is orthogonal to the raindrop axis. There is also an ambiguity in definition of quantities such as Linear Depolarisation Ratio. Authors such as Bringi and Chandrasekar, refer to LDR as being the value of the Linear Depolarisation Ratio for the case when the radar beam is orthogonal to the drop axis – indeed we believe that this is the generally accepted terminology. Others, however, (e.g. RAL) refer to LDR as being the linear depolarisation ratio at any angle (in particular for a vertically pointing radar). This leads to confusion, and the reader is urged to ensure that he/she checks the definitions of the parameters in any particular paper/article. The second point to bear in mind is that some of the radar parameters are direct measurables in one polarisation basis, whereas they are only derived from measurements in another polarisation basis. In either basis, the data may also be a convolution of the intrinsic back-scatter parameters and the on-path propagation effects. For example, Differential Reflectivity ZDR can be measured directly in a linearly polarised system, whereas in a circularly polarised system it may be derived from the measurements (ref. 35), but the effects of propagation are worse (ref. 6). In a hybrid (ref.36) polarisation system in which 45 linear polarisation is transmitted, the vertical and horizontal components of the return field are measured. ZDR may be directly obtained from the measurements (which do not require switched transmission), but the propagation effects are different from when alternate switched transmission is used (ref. 37, §4.7). Similarly, differential propagation phase KDP may be derived from alternate H,V measurements using a linearly polarised Doppler radar, or it can be obtained from a simultaneous measurement using a (non-coherent) circularly polarised radar which measures both received powers (left-hand and right-hand circular {LHC and RHC}), and the complex correlation of the LHC, RHC fields. The advantage of the latter is that there is no ambiguity in phase in the non-Doppler system, and the element of noise brought into the linear measurement due to the necessity of using switched transmission should be absent. Neither measurement is subject to propagation effects, but the difficulty with both is the accuracy available in the phase measurement. This is discussed in §5 . The co-polar linear correlation coefficient is also not affected by propagation effects, but it is only available directly to a linearly polarised radar. However, it is closely related t the “degree of polarisation” which can be obtained in a circularly polarised system. The Circular Depolarisation Ratio is simply the ratio of the received powers in the LHC and RHC fields when just one field is transmitted. For spherical raindrops, therefore, CDR (on a dB scale) is large and negative being, in fact, dominated by the antenna characteristics. ORTT is a parameter which depends both on the nature of the size distribution of the particles in the resolution volume, and on their orientation. |
, one can construct the Hermitian coherency matrix (refs. 23, 2 ),
expresses the intensity in the wave ensemble, while the remaining components may be regarded as forming a 3-dimensional vector. In the case of a fully polarized wave there is a perfect correlation between the cartesian components of the electric field. This implies that the rows and columns of the coherency matrix are not linearly independent, so that the determinant vanishes. Hence there is then a relation between the components of the Stokes vector for a coherent