Linear correction

The linear correction methodology for attenuation and differential attenuation is derived from the proportionality of attenuation (both Ah and ADP) to differential phase shift

(KDP), as shown in Equation 6.3 (Bringi and Chandrasekar, 2001). The parameter α is

a function of the radar frequency, temperature and the atmospheric conditions and has been shown to vary in time and space particularly at C-band and X-band frequencies (Tabary et al., 2009; Park et al., 2005).

Ah =αKDP(r) (6.3)

As the attenuation at any point along a radial is a function of all preceding attenuation, a correction for attenuation can be achieved using a range integral (Equation 6.4), where Zcis the corrected reflectivity andZm is the measured reflectivity. By then substituting

the specific differential phase into Equation 6.4 using the relationship shown in Equation 6.3 (Equation 6.5) and expanding the integral of specific differential phase shift along a path to simply the change in path forward differential phase shift (ΦDP) along the path

10 log₁₀[Zc(r)] = 10 log10[Zm(r)] + 2 r Z 0 Ah(s)ds (6.4) 10 log₁₀[Zc(r)] = 10 log10[Zm(r)] + 2α r Z 0 KDP(s)ds (6.5) 10 log₁₀[Zc(r)] = 10 log10[Zm(r)] +α[ΦDP(r)−ΦDP(0)] (6.6)

Equation 6.6 can therefore be applied to correct for attenuation, using the smoothed profiles of forward phase shift calculated in the previous section, provided a value forαcan be defined. Park et al. (2005) show that α varies between 0.139 and 0.335 dB/deg at X- band with a mean value of 0.254 dB/deg in their scattering simulations, while A. Ryzhkov (personal communication, April 2015) expects α to vary between 0.17 and 0.35 dB/deg with an expected value of 0.27 dB/deg. Similarly differential reflectivity can be corrected in the same way by exchanging the proportionality constant α with the proportionality constant between differential attenuation and specific differential phase (β). Again β can take a range of values, typically in the range 0.03 to 0.06 dB/deg as noted by A. Ryzhkov (personal communication, April 2015). Due to the inherent variability of these parameters, and the simplification in Equation 6.3 (omitting the exponent of KDP to

provide linearity), the correction obtained from this approach is only approximate, but is significantly more stable than the methods of attenuation correction typically applied to single polarisation radar data (see Section 2.2.2).

The linear method has been implemented for both a stratiform and convective rainfall event during the COPE campaign to identify the uncertainty introduced by the method. For the stratiform case with widespread rainfall the uncertainty increases with distance from the radar as the phase shift slowly increases with range, with every 5 degrees of phase shift introducing an uncertainty of 1 dBZ for corrected reflectivity and 0.15 dB for differential reflectivity if taking the widest range of values possible for α and β.

Figure 6.3. demonstrates the range of uncertainty introduced through attenuation correc- tion for both horizontal reflectivity and differential reflectivity during a stratifrom rainfall event with low differential phase shift (<20◦). The phase shift profile used to correct the reflectivity measurements is shown in the bottom panel, and shows periods of decreasing

Chapter 6. Correction using dual polarisation 107

Figure 6.3: Linear correction of horizontal reflectivity and differential reflectivity using a linear transformation of ΨDP. Data shown is taken from the 2013-08-17 12:16UTC

radar volume at an azimuth of 72◦and elevation of 0.5◦. Measured radar variables are shown by solid red lines, with the top panel showing horizontal reflectivity corrected with anαof 0.27 dB/deg (solid black line) and bounded withαof 0.14 and 0.35 dB/deg, the middle panel showing differential reflectivity corrected with aβof 0.045 dB/deg and

phase shift which are a result of noise smoothing rather than a true atmospheric signal. During convective events the smoothing no longer exhibits these fluctuations as the atmo- spheric phase shift dominates the noise. Figure 6.4 shows a single radial from 2 August 2013 with almost 100◦of atmospheric forward phase shift as a result of an evolving con- vective line system. Both the measured profile of horizontal reflectivity and differential reflectivity show significant attenuation, with differential reflectivity of -6 dB in rainfall being over 6 dB less than the expected value in rainfall. Smyth and Illingworth (1998a) previously used the positive differential reflectivity signal in the rainfall region beyond an attenuating cell to constrain the value of β within the cell, which would indicate that even the expected value of 0.045 dB/deg is too low for the atmospheric conditions shown in Figure 6.4 as the corrected value (solid black line) descends to below 0 dB in this region. The potential for under correction of differential reflectivity is significant in these convective events where high differential phase shift is observed. This uncertainty should be considered when using differential reflectivity for rainfall estimation, as has been covered in Chapter 7.

The following section presents the ZPHI methodology which distributes attenuation with the measured reflectivity constrained by the total phase shift along a radial. The advan- tages of this methodology are the removal of errors resulting from smoothing fluctuations in the phase shift or the presence of backscatter differential phase and also that the ZPHI algorithms return the specific attenuation for each range gate, which can be utilised for rainfall estimation and radar correction.