Rain drop shapes play an important and a central role in evaluating radiowave propagation effects [1,2] for earthspace and terrestrial links as well as rainfall remote sensing by polarimetric radars [3,4,5]. In the former case, information on drop shapes are needed for evaluating rain attenuation effects and the polarization dependence as well as rain-induced depolarization effects, often expressed in terms of the variation of the cross-polar discrimination (XPD) with the co-polar attenuation (CPA) [6,7]. For the latter case, they are needed for estimating rainfall rates and microphysical parameters more accurately [8] which have applications in hydrology and meteorological modelling respectively.
Our understanding of drop shapes for drop diameters > 2 mm is now on a firm footing from both precise wind-tunnel measurements [9] and from 2D-video disdrometer (2DVD) measurements [10] from the "80-m fall experiment" as reviewed in [11]. The most probable shapes derived from 2DVD in [10] have been shown to be in excellent agreement with the equilibrium numerical model of [12] well within their estimated error bounds. 2DVD measurements in natural rain have also been made but only a few cases had simultaneous observations from polarimetric radar, for example [13,14].
In this paper, we examine 2DVD measurements during a category-1 Hurricane (Dorian) event whose rain-bands had traversed a ground instrumentation site located in Delmarva peninsula, USA. An S-band polarimetric radar, 38 km from the 2DVD site, had been used to perform continuous observations. The scan sequence included range-height indicator (RHI) scans over the disdrometer site.
Scattering calculations are performed on a drop-by-drop basis, and subsequently used to determine the S-band copolar reflectivity (ZH) and differential reflectivity (ZDR) which in turn are compared with the radar observations. We draw an important conclusion from these comparisons.
Hurricane Dorian made landfall in the United States near the Florida coastline at the end of August 2019. It then moved along the east coast of the continent, later becoming a post-tropical cyclone. The rain-bands of the storm had traversed a site at the NASA Wallops Precipitation Research Facility [15] where a network of ground instruments including several 2DVDs had been installed. The site is within the coverage of NASA's S-band polarimetric radar, NPOL.
The event which occurred on 9 September 2019 lasted for more than 8 hours [16] and here we analyze 2DVD measurements and radar observations from 11:00 UTC to 19:00 UTC. The event had unusually large drops. Fig. 1 shows the 3-minute drop size distribution (DSD) for the 8hour period. The color scale represents the drop concentration and the thick grey line represents the maximum recorded equi-volume drop diameter within each 3-minute interval. As can be seen drops larger than 5 mm were recorded several times and the largest drop that was recorded was 8.06 mm at 15:58 UTC. A study using microphysical simulations [17] together with radar observations of vertical profiles (especially of co-polar correlation coefficient, ρhv) over the 2DVD location had clearly indicated that these were all fully melted hydrometeors (i.e. rain drops) at ground level. An example of an RHI scan of ZH taken at 16:02 UTC is shown in Fig. 2(a) and the vertical profiles of ZH, ZDR and ρHV over the 2DVD are shown in Fig. 2(b), (c), and (d) respectively. Despite the presence of large drops at ground level, the RHI shows that it was stratiform rain as indicated by the radar bright-band seen clearly between 3.5 to 4 km height above ground level (representing the melting layer region). The vertical profile in panel (b) shows > 40 dBZ near ground level whereas panel (c) shows that the corresponding ZDR is only about 1 dB. ρHV in panel (d) is very close to 1 below the melting layer down to ground level implying fully melted hydrometeors, especially close to ground. Note, due to beam blockage, radar data below 500 m (above ground level) need to be omitted from further consideration.
To reconstruct the drop shapes from the 2DVD images, we make use of the recorded contours of each drop in the two orthogonal planes. The procedure has been described previously [18,19,20] hence will not be repeated here, but one point to note is that, owing to a number of limitations, the shape reconstruction was only performed for relatively large drops, viz. for equi-volume drop diameter (Deq) > 2 mm. An example of a reconstructed large drop is shown in Figure 3. Note the drop does not possess rotational symmetry. For smaller drops, we use the most probable shapes with rotational symmetry from [10].
The scattering calculation of each reconstructed drop has been carried out, using CST Studio Suite 2020, a 3D electromagnetic (EM) analysis software package that provides EM solvers for application across a wide frequency spectrum. For the required calculation of the radar cross section (RCS) of the raindrops at 2.8 GHz frequency, the built-in Integral Equation Solver has been used. The scattering calculations have been automatized by controlling CST via an application programming interface (API), scripted by Matlab code.
Fig. 4 shows an example of the reconstructed drop from Fig. 3, after importing into CST and after performing a surface triangulation. As material, the dielectric properties of water at 2.8 GHz frequency at a temperature of 20° C have been calculated by applying the formulae of Ray [21]. Fig. 5 shows the simulated RCS of the example drop for both horizontal and vertical polarization, as a function of the horizontal view angle. Note for an equi-volume sphere this value would be -72.3 dBm² for all angles. Here we use the RCS values corresponding to the view angle from the radar.
WITH RADAR DATA RCS values for H and V polarizations for each drop were used to compute ZH and ZDR over every 3 minute interval from 11:00 to 19:00 UTC using the same approach given in [13,14]. For around 5% of a total of 15202 drops > 2 mm, the reconstruction was not possible. For those drops (and for the drops ≤ 2 mm) the RCS values corresponding to the most-probable shapes were used. The radar reflectivity is calculated from the individual scattering amplitudes, for example over a 3-minute period, by performing drop-by-drop integration of the radar cross-sections (in actual fact the covariance matrix elements) during the specified time period. If the H-polarization reflectivity for the i th drop is denoted by zi h , then the overall reflectivity from all drops over the 3minute time interval is:
where vi is the vertical (fall) velocity of the i th drop, A represents the measurement area of the 2DVD, and, Δt represents the averaging time period which in this case is 180 seconds. For V polarization, similar integration is performed using the corresponding RCS values, zi v . Both are converted to the conventional dBZ units and ZDR for each of the 3minute period is determined from the difference between the two.
The computed ZH and ZDR are shown as green points in Panels (a) and (b) of Fig. 6. These are compared with the NPOL radar data extracted over the 2DVD site shown as black lines. Because of the aforementioned beam blockage, data at a height of 500 meters have been used, and moreover, 3 pixels on either side (= 450 m) of the 2DVD site are included. Additionally, for ZDR, a previously reported filtering technique was applied [16]. The unfiltered ZDR are shown as grey points. Fig. 7 also shows the ZH and ZDR calculated using 3minute DSDs assuming the most-probable (= equilibrium) shapes, represented by the orange points. Apart from a 9 minute time interval around 16:15 UTC, the green line shows better ZDR agreement than the orange points, the latter tending to be higher than the ZDR from NPOL. On the other hand, ZH does not show any noticeable differences. Note radar calibration of ZH and ZDR was extensively performed throughout the event.
In Fig. 7, we show the variation of (unfiltered) ZDR versus ZH from NPOL (grey points) as well as those from the two sets of scattering calculations. They are all the same points as in Fig. 6. Also shown, as purple line, is the powerlaw fitted curve to the radar data. The better agreement with the green points is seen more clearly. The calculated relative bias (RB) had a mean of 1.9% for the green points and -13% for the orange points and their corresponding standard deviations were 43% and 54%. The frequency of occurrence of ZDR versus ZH derived from 9 RHI scans are shown in Figure 8 together with the same scattering calculations using the two methods. The drop-by-drop calculations are once again seen to be in better agreement with the radar data. V. DISCUSSION AND CONCLUSIONS
Comparison of RB values relating Fig. 7, together with the comparisons presented in Fig. 8, clearly show that the scattering calculations using individual drop shapes are significantly closer to the (calibrated) radar data than those using the equilibrium or the most probable shapes. The latter overestimates the ZDR considerably. The main implication is that for this Hurricane (rain-bands) event, a significant fraction of the (> 2 mm) drops deviate from the most probable shape, tending towards sphericity. Large amplitude mixed mode oscillations are very likely to be responsible, in particular transverse oscillation, mode (2,1). The lower than expected ZDR values e.g. for Z > 35 dBZ from the NPOL data is also consistent with previous radar observations at Sband for other hurricane events [22]. One factor was thought to be to the presence of high concentration of small drops as has been previously measured in such storms [23,24]. On the other hand, the ZDR calculations shown as orange points in Figure 6(b) are based on the 'full' DSD spectra, utilizing not only 2DVD data but also data from a collocated Meteorological Particle Spectrometer [25,26] capable of measuring drop concentrations down to 0.2 mm. In [17] where output from a particle-based microphysical method is used to compare with radar observations for this very event, the authors noted the "unknown effects of strong wind gusts of a category-1 hurricane near the surface in disturbing the most probable shapes and orientation (canting angle) of drops deduced from the 80-m fall bridge experiments", the latter being given in [10]. Furthermore, in [17], the standard deviation of the effective canting angle needed to be as high as 20° (much higher than the conventional assumption of 7-10°) "to account effectively for multi-mode oscillations due to the strong wind gusts and turbulence". In support of these points, we show in Fig. 9 the variation of single particle ZDR calculated using the reconstructed drop shapes with Deq for all drops > 2 mm. They are represented by the green dots. The red lines represent the ± one standard deviation (σ) for each Deq. As can be seen, σ is rather large, and in fact increase with Deq. The '+' points in blue represent the ZDR calculations using the most probable shapes from [10].Whilst they lie within the ±σ lines, they seem closer to the upper red line.
Another interesting point to note is that ±σ increase with increasing Deq implying that the drop oscillation amplitudes increase with Deq. This increase has indeed been noted with both wind-tunnel observations as well as the 80-m fall experiment, for example, see [27].
Thus, such storms (often associated with significant gust) may require modified polarimetric radar retrieval algorithms for estimating rainfall rates compared with more commonly occurring rain events. The estimators will need to take into consideration the change in the overall ZDR, say within a radar pulse volume, occurring as a result of large amplitude mixed-mode oscillations. (Future work will address this important issue.) The 2DVD provides important drop shape information pertinent for such applications, as has been demonstrated here. Note also that in a previous study [13] relating to rain bands of a Tropical Depression (Nate), the overall ZDR was not found to vary systematically from most other events. The winds associated with the Nate event were significantly lower than the hurricane event considered here. Finally, as mentioned in the Introduction, drop shapes also have an impact on the XPD versus CPA variation needed for earth-space communication links [6,7]. Data from 2DVD have been used in prior studies to simulate beacon experimental scenario and compare with actual measurements. For example in [28], simulations were carried out for the 20 GHz band and compared with beacon measurements taken in Aveiro, Portugal, over a 1-year period. Rain drop size distributions as well as the most probable shapes from [10] were used. In a later study [29], the spread in the XPD-CPA was illustrated when individual drop shape information was included. However, the individual drop shape was defined in terms of effective axis ratios (and assuming rotationally symmetry) as well as the individual orientation angle. These studies can be improved by utilizing the 3D reconstructed shapes of individual drops as well as the 3D electromagnetic analysis software packages which are now available. Note however, the XPD-CPA calculations will require complex forward scattering amplitudes for individual drops in addition to the complex backscatter amplitudes.