3.3.1. Bias Assessment
The gain and loss differences between the horizontal and vertical channel induce systematic bias in the
estimate. The accuracy of
with 0.2 dB is required for the application of QPE or hydrometeor classification [
17] and self-consistency process, therefore, careful absolute calibration is necessary. The two methods for
bias correction are employed here considering the respective strengths and limitations.
Light rain at low angle (LRLA). The shape of large-sized free-falling rain drops are modeled as non-spherical oblate spheroids [
22]. This is the result of forces and surface tension acting around the drops. Moreover, rain drops with a diameter size less than 0.5 mm can be modeled using a nearly spherical shape. This inherent microphysical property of small drops can be used to estimate the
bias. Due to the spherical shape of small drops (raindrop axis ratio ≈ 1), the power return from both polarizations (horizontal and vertical) is expected to be identical. This will lead to an expected measured mean
of approximately 0 dB plus/minus the estimated radar measurement bias. The mean
was estimated only using the pixels with
greater than 0.95, SNR greater than 10 dB
ranging from 10 to 15 dBZ. Besides, rainfall over the radome can induce signal attenuation and lead to the uncertainty of bias estimation. The data at times without rain over the radar radome were selected for assessing the systematic bias.
Figure 6 shows an example scatter of
versus
captured at 1.5° elevation angle of one rainfall event at 16:25 (BJT) on 4 June 2016. It can be seen from the plot that the mean
increases exponentially with
. For this case, the average bias is 0.481 dB.
Dry aggregated snow (DAS). The average
values of aggregated snow normally do not exceed 0.25 dB and tend to slowly decrease with increasing
[
17]. Considering the low variability of the expected power returns from dry aggregated snowflakes between the S- and X-band, the estimated value of 0.2 dB accounts for the absolute calibration of
at X-band [
18]. Dry aggregated snowflakes are universally present above the melting layer in stratiform clouds. The existence and identification of bright bands becomes the prerequisite for calibration using ‘dry aggregated snow’ method. A number of polarimetric observations show that the aggregated snow likely occurs around 1–2 km above the bright band [
27].
Figure 7 represents a vertical profile example case of
,
, and
from the stratiform precipitaon at 16:25 (BJT) on 4 June 2016.
is the main polarized variable which can be used to identify the melting layer and freezing-level heights, and discriminating among rain, snow, and melting-level regions. The magnitude of
is generally in the range of 0.7 to 0.95 in the melting layer. For this case, a
lower than 0.8 was observed at the melting layer where the height is 3.3 km. Upon this, the average value
at the height between 4.3 and 5.3 km was approximate to the systematic bias, which is about 0.52 dB and nearly equal to that estimated by the ‘light rain at low angle’ approach for the same case.
Two approaches were applied for the whole rainfall event to estimate daily
bias. The ‘light rain at low angle’ approach was eligible for all of the rainfall events. The ‘dry aggregated snow’ approach was performed only for the stratiform rainfall event, considering that dry snow is hardly identified for convective rainfall. There were six stratiform rainfall processes for fifteen rainfall events.
Table 4 shows the mean and standard deviation of the
bias estimated by the DAS and LRLA approaches for all rainfall events.
Several characteristic of
bias over IOP can be found from
Table 4, such as: (1) The mean of the
bias estimated by the ‘dry aggregated snow’ approach and the ‘light rain at low angle’ approach varied from 0.52 dB to 0.79 dB, and from 0.43 dB to 0.81dB, respectively, The standard deviation of
bias estimated by the ‘dry aggregated snow’ approach and the ‘light rain at low angle’ approach varied from 0.13 dB to 0.22 dB, and from 0.17 to 0.32 dB, respectively; (2) The changing trend of
bias can be seen as slightly increasing from both approaches.; (3) The ‘dry aggregated snow’ approach has a lower estimate standard deviation than the ‘light rain at low angle’ approach; (4) The overall average of
bias is 0.68 dB and 0.65 dB, respectively, for the DAS and LRLA approach.
Based on the quantitative bias assessment, the biased
can be corrected as:
where
is measured differential reflectivity,
is averaged bias for each event, and bias correction was performed with separate
for each rainfall event.
3.3.2. Bias Assessment
The self-consistency principle claim that the polarimetric variables of
,
and
lie in a limited 3-D space for rain medium [
20]. As such,
measurements can be reconstructed from
and
measurements, as defined below.
where
represents the specific differential phase reconstructed from
and
,
and
are in linear units. Before reconstruction,
is corrected for attenuation and bias, and
is corrected for attenuation. Two methods were evaluated for
bias correction: the intrinsic properties of dry aggregated snow present above the melting layer, and light rain measurements close to the ground. The parameters
a,
b, and
c depend on the size, shape, and distribution of raindrops and can be calculated using rain simulations with a gamma DSD and a fixed drop axis ratio relationship. The bias
in
can be obtained using the following relationship:
where
is the computed specific differential phase obtained from the measured radar differential phase.
The self-consistency principle was applied for the fifteen rainfall events. Based on the radar variables simulated from the in-site raindrop spectra data, the parameters
a,
b, and
c were regressed as 2.22 × 10
−4, 1, and −4.58, respectively, shown in It is worthwhile noting that the exponent
and
are close to unity, and the linearity has a good approximation at frequencies from 2.8 to 9.3 GHz [
25,
26]. Due to the variation of raindrop shape, the coefficient
varies from 0.139 to 0.335 dB/deg and the coefficient
also varies from 0.114 to 0.174 dB/deg. The variation differences of
or
influenced by temperature is much smaller than that of coefficient
or
. The uncertainty of
and
accounts for 28% or 17% relative errors to the mean value respectively [
12]. The simulations for
and
were performed as shown in the scatterplot of
Figure 3. Through nonlinear regression processing,
and
here are 0.323 and 0.131,
and
are 1.05 and 1.2, as shown in
Table 3. The correlation coefficient between
and
,
and
is 0.99 and 0.96, respectively, and indicates that the empirical relations are eligible for attenuation correction at the X band, based on the self-consistent method.
With the attenuation and bias corrected
and retrieved
, the
bias assessment using the self-consistency principle was performed for rainfall events. In order to validate the efficiency of self-consistency-based bias assessment, it was performed to compare XPRAD and the two closest S-band CINRAD radars (9200 and 9762) that were supposed to be well-calibrated. To perform the X-band and S-band data comparison, scatter plots of
(
from S-band) versus
(
from X-band) were generated for the same radial direction at the elevation of 0.5°, shown as
Figure 8a,b. Terrain height changing between the X- and S-band radar is also considered with the usage of the DEM (digital elevation model) data, shown as green color padding in
Figure 8a,b. There is some slight terrain blockage for CINRAD (9200) at the azimuth of 35°. The beam blockage ratio (BBR) was calculated with the method described in [
28]. The BBR along this radial varies from 0.03 to 0.24, and the reflectivity was compensated with the BBR calculation.
For radar data selection, only 0.5° PPI scans with a data collection time difference of less than 1 min were selected. Pixels where both the S- and X-band data coexist within a
value in the range of 15 dBZ to 45 dBZ were selected to limit the errors due to uncertainty in low
returns and the deviation from Rayleigh scattering of big drops from high
returns in the X-band data.
Figure 8a,b shows the common radial coverage area at the elevation of 0.5° between the S-band radar and the X-band radar along the same radial direction. The S-band range bin length was 250 m, the X-band range bin length was 75 m. The common range bin step was 750 m. Within the common radial range, the reflectivity at every 750 m interval for the common coverage of both radar was selected. After the data pixel selection was performed, the mean
and mean
values were computed for each radar separately for all the available reflectivity values from the PPI scan at the elevation of 0.5°.
To estimate the reflectivity bias between the CINRAD and XPRAD, the following statistical approach was used:
where
n represents the number of
meeting the selection condition for each radar throughout each rainfall event,
is the reflectivity from XPRAD and just corrected for attenuation prior to bias correction with self-consistency processing,
is the reflectivity from CINRAD.
The
bias estimated by the self-consistency approach and the two closest CINRAD are shown in
Table 5. There are no comparison outcomes between CINRAD and XPRAD for rainfall event number 4, 10, 11 and 14, since the corresponding rainfall event did not occur at the common radial coverage. The
bias estimated by the self-consistency approach, CINRAD (9200) and CINRAD (9762) varies from 0.10 dBZ to 1.38 dBZ, from 0.11 dBZ to 1.23 dBZ, and from 0.15 dBZ to 1.41 dBZ, respectively. The overall average
bias estimated by the self-consistency approach, CINRAD (9200) and CINRAD (9762) is 0.60 dBZ, 0.65 dBZ, and 0.75 dBZ, respectively. The maximum
bias estimated is 1.41 dBZ, less than 1.5 dBZ. The maximum standard deviation of
bias estimated is 0.48 dBZ, less than 0.5 dBZ. Such small differences demonstrate the feasibility of the application of the self-consistency criterion in a dual polarization radar measurement quality check.
Based on the quantitative assessment, the biased
can be corrected with a self-consistency estimation, as:
where
is the measured reflectivity,
is the averaged bias for each event, and bias correction was performed with a separate
for each rainfall event.