As already mentioned, the addition of a second, orthogonal, plane of transmission to weather radars allows the measurement and estimation of several new radar moments based on differences between the two planes. These new moments provide additional information about the echoes being sampled, which are of great use in radar hydrome- teorology. These new parameters are the differential reflectivity (ZDR), the differential
phase shift (ΨDP) (which is composed of the forward propagation phase shift (ΦDP) and
the backscatter differential phase shift (δco)), the specific differential phase (KDP), the
co-polar cross correlation (ρco) and the linear depolarisation ratio (LDR). The follow-
ing section describes each of these parameters in turn, along with a description of their physical meaning. The benefits of these new moments are then covered in the following section (2.4)
2.3.2.1 Differential reflectivity
Differential reflectivity (ZDR) is the observed ratio of the horizontally and vertically polarized linear reflectivity measurements. It was first proposed as a method of ob- serving rainfall by Seliga and Bringi (1976), with the aim being to add information to help quantify the rainfall drop size distribution. Due to the oblate spheroidal shape of falling raindrops, they produce a positive reflectivity shift in the horizontal, relative to the vertical, which is proportional to their diameter. Scattering simulations and field measurements show that ZDR ranges from 0.2 dB in very light drizzle to over 4.5 dB for very large rain drops (Seliga and Bringi, 1978; Hall et al., 1984; Balakrishnan and Zrnić, 1990, for example). Scattering simulations by Ryzhkov and Zrnić (2005) quantified this relationship for S, C and X-band wavelengths and show the variation between them, with X-band suffering from minor resonance effects at a diameter of 3.5 mm and C-band suffering from more extreme resonance effects at diameters above 5 mm (Fig. 2.5). These resonance effects are the result of Mie scattering (Matrosov et al., 2002; Meischner et al., 1991), with the reduced resonance at X-band believed to be a consequence of increased absorption dampening the resonance effect (Park et al., 2005; Ryzhkov and Zrnić, 2005).
0 1 2 3 4 5 6 7 8 Equivolume diameter (mm) 0 1 2 3 4 5 6 7 8 ZD R (d B ) X-band S-band C-band
Figure 2.5: Theoretical variation of differential reflectivity with increasing equivolume
Chapter 2. Weather radar for hydrology 21
2.3.2.2 Differential phase shift
The total differential phase shift (ΨDP) is the observed phase difference between the received horizontal and vertical pulses and contains two components, the backscatter differential phase (δco) and the propagation differential phase ΦDP (Eq. 2.8) (Bringi and
Chandrasekar, 2001). As there is often a phase shift on transmission the received phase shift also contains this component, which must be removed to gain the shift resulting from atmospheric interaction.
ΨDP = ΦDP +δco (2.8)
The differential propagation phase measures the change in phase between the horizontal and vertical phase of returns along the beam path which varies as a function of the total cross section of scatterers along the path. Therefore it varies as a function of both axis ratio of the drops and number of drops, with measurements in rainfall monotonically increasing along the beam path due to the positive axis ratio of raindrops (Seliga and Bringi, 1978; Jameson, 1985). Due to this dependence on the DSD, phase shift has frequently been proposed as an additional means of estimating rainfall using weather radar (Sachidananda and Zrnic, 1986, for example). These approaches have often been limited to higher rainfall rates as the phase shift is inversely proportional to wavelength for a given rainfall rate, with most early studies using S-band systems (Matrosov et al., 1999). Conversely the phase shift for an X-band system is 3 times greater than at S- band, which increases the sensitivity to lower rainfall rates. As a cumulative quantity the propagation phase is often differentiated with range to calculate the specific differential phase (KDP) for each range gate for the purposes of rainfall estimation (see next section).
The backscatter differential phase forms the second component of the observed phase difference (Eq. 2.8). It is a function of non-Rayleigh scattering and is not cumulative along the rain path but a function of the distribution of scatterers in each range gate. Due to it being a function of non-Rayleigh scattering it is often ignored at S-band, but becomes an increasing component of the measurements at lower wavelengths, particularly X-band when equivolume drop diameters exceed 2 mm (Matrosov et al., 1999). As the backscatter phase has often proven difficult to separate from the total phase difference
measurements its use has been limited, however recent studies including Trömel et al. (2013) and Tyynelä et al. (2014) have begun to explore the potential of using backscatter differential phase, particularly for observing melting particles.
2.3.2.3 Specific differential phase
The specific differential phase (KDP) is simply the range derivative of the forward prop-
agation differential phase ΦDP, typically expressed in units of degrees per kilometre (◦ km−1). It is a calculated, rather than measured, radar moment, being derived from one component of the measured differential phase shift. At shorter wavelengths calcu- lation requires either the filtering of the observed phase shift to remove the backscatter component, or careful selection of the calculation path to ensure the change in backscat- ter between the start and end of the path is minimal. The simplest method of calculation is a finite difference along a range path, however for rainfall estimation a more accurate method is required (Bringi and Chandrasekar, 2001). These methods include linear re- gression over varying path lengths (Ryzhkov and Zrnic, 1996), iterative range smoothing (Hubbert and Bringi, 1995), solving linear equations with linear programming (Gian- grande et al., 2013) and even calculation using the angular domain with cubic spline estimation of KDP (Wang and Chandrasekar, 2009).
2.3.2.4 Co-polar cross correlation
The co-polar cross correlation coefficient (ρco) is the correlation between the co-polar
return at horizontal polarisation and the co-polar return at vertical polarisation. For alternating transmission mode only one of the co-polar elements can be measured per pulse and the co-polar correlation has to be estimated given the time lag between pulses. For simultaneous transmission the cross-polar components of the received powers are negligible compared to the co-polar components of the powers and therefore the co-polar cross correlation can be taken to be the measured co-polar correlation. However, in the presence of cross-polarizing scatterers, the measured value begins to deviate from its intended physical description as detailed by Galletti and Zrnić (2012). By measuring the correlation between the two pulses the homogeneity of the sample volume can be assessed, for example raindrops exhibit a large degree of spatial homogeneity and therefore have
Chapter 2. Weather radar for hydrology 23
correlations above approximately 0.97 (Balakrishnan and Zrnić, 1990). Mixed phase returns, particularly the melting layer have lower correlations, typically reported to be around 0.9 (Ryzhkov et al., 2005b; Bringi et al., 1991; Balakrishnan and Zrnić, 1990), although this is somewhat dependant on the radar configuration (particularly the dwell time and the beamwidth), for example Illingworth and Caylor (1989) observed values as low as 0.6 in the bright band with a 0.25 degree beamwidth S-band radar, while Zrnić et al. (2006) showed the variation of correlation with antenna rotation speed, with faster speeds leading to lower expected values. As a result the co-polar cross correlation is a useful parameter for identifying mixed phase returns, and also for identifying regions of non-meteorological echoes (see 2.4.1).
2.3.2.5 Linear depolarisation ratio
The linear depolarisation ratio is the amount of depolarisation which occurs due to cross polarising scatterers. To measure LDR requires the radar to operate in alternating trans- mission mode, such that a single polarisation is transmitted at any one time, and that both received polarisations are recorded. The LDR is then the amount of signal that is converted from the transmitted orientation (either horizontal or vertical) to the or- thogonal direction by the observed hydrometeors. LDR increases with rainfall intensity as heavier rainfall has a more varied DSD, but is greater still for melting snow, hail, and strongly orientated ice crystals which all exhibit more variability in their shape and movement during descent (Bringi et al., 1986; Brandes and Ikeda, 2004; Ryzhkov and Zrnic, 2007). As such LDR is most often used in identification of scatterers, particularly solid phase scatterers.