2.3. Attenuation
2.3.2. Constrained Forward Correction
reflectivities only. They provided several algorithms based on different amounts of knowledge of either PIA, initial attenuation (e.g. by a radome) or no additional information. They showed that the algorithm using no additional information would become unstable in the same way that the HB method would. However, if a reference for the PIA was available, their algorithm would become more and more stable and accurate with increasing precipitation intensity as opposed to the HB algorithm.
Delrieu et al. (1997) tested and applied (Delrieu et al., 1999b,a, 2000) a method to use ground clutter returns to get an estimate of PIA by means of radar data only.
Their approach uses two scans at different elevations to remove any signal caused by rain above the clutter bin. In addition it relies on data, which is unaltered e.g. by a Doppler filter as presented in section 2.2.1.
For the case when a PIA reference is not available, Nicol and Austin (2003) sug-gested the introduction of an adjustment parameter q, which could be tuned so that the attenuation correction would remain stable and a certain PIA would be achieved.
One problem for application of the Marzoug algorithms is, that they use aZ-k re-lation instead of ak-Z relation. For these relations, parameters must be determined either empirically or using backscattering models based on certain assumptions on the drop size distribution. These parameters are dependent on the wavelength and were calculated for X-Band and shorter wavelengths only, as these bands are pre-dominantly used in satellite radar systems. As the radar data available for this study comes exclusively from C-band systems, theZ-kparameters needed to be determined before this method could be used for attenuation corrections at the 5 cm wavelength.
Whilek-Z parameters can be found in the literature (e.g. in Kr¨amer (2008) or Battan (1973)), due to their empirical nature and the poor log-log linearity of the relation (Delrieu et al., 1999a), these cannot simply be inverted without introducing large errors.
2.3. Attenuation
In short, the Kr¨amer algorithm is based on a k-Z relation of the form
k=αZβ (2.16)
which giveskin [dB/km] whileZ needs to be given in linear units of [mm6/m3]. The recursive correction formula, calculating the actual reflectivityZi from the apparent (i.e. measured) reflectivity Za,iat the range bini is then
Zi = Za,i·
i−1
Y
j=0
10kj/10
= Za,i·
i−1
Y
j=0
10
αZjβ
/10 (2.17)
For numerical reasons, these calculations are usually done in decibel(dB)-space, which also simplifies the product to a summation
dBZi =dBZa,i+
i−1
X
j=0
α
10dBZj/10β
(2.18) Fig. 2.8 shows the effect of the attenuation correction on a long term accumulation.
As expected the amounts are increased everywhere. Most notable is the increase towards the south where the Allg¨au Alps are located. According to Frei and Sch¨ar (1998) this area should provide an annual total between 4 and 5 mm/day, which would amount to a total accumulation between 1460 and 1825 mm, which is more realistically approached by the attenuation corrected accumulation. Some of the residual clutter, including the spoke to the southeast, however also gets increased.
Although this shows the importance of proper clutter filtering, on the other hand, a radial accumulation bias due to an intensive clutter pixel is not seen in the figure.
Fig. 2.9 presents differences and ratios between the accumulations displayed in figs. 2.8a and 2.8b. While it can be seen in fig. 2.9a that the maximum accumulated correction amounts to approximately 300 mm for the year 2008, fig. 2.9b shows that the relative change increases radially, as expected, reaching its maximum in the regions of lower total accumulations in the northern semicircle, where the average correction is about 20-30%.
Both figures show that the two constraints are apparently effective in preventing gross overestimation while also, on average, limiting the amount of correction to reasonable values.
Another comparison was made on the hourly timescale. For this purpose hourly data from a set of ca. 340 operational gauging stations provided by the DWD were compared with hourly accumulations of clutter and attenuation corrected radar data for a period of 3 months between May and July 2008. This period comprises the event on 2nd of June, which led to devastating flash floods in the Starzel catchment (Ruiz-Villanueva et al., 2012). The stations were filtered to remove those which only sporadically provided data. It was found that 53 stations provided values for more than 95% of the time, while availability drastically decreased afterwards. In order to remove possible spurious correlations, the analysis was limited to these 53 stations.
0°
45°
90°
135°
180°
225°
270°
315°
0 150 300 450 600 750 900 1050 1200 1350
(mm)
(a) uncorrected
0°
45°
90°
135°
180°
225°
270°
315°
0 150 300 450 600 750 900 1050 1200 1350
(mm)
(b) corrected
Figure 2.8.: Effect of attenuation correction, radar T¨urkheim. (a) Accumulation for year 2008 corrected for clutter only, (b) clutter and attenuation corrected accumu-lation
0°
45°
90°
135°
180°
225°
270°
315°
0 30 60 90 120 150 180 210 240 270
(mm)
(a)difference
0°
45°
90°
135°
180°
225°
270°
315°
1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
(-)
(b) ratio
Figure 2.9.: Relative effects of attenuation correction, radar T¨urkheim, accumulation year 2008. (a) Difference between corrected and uncorrected (c.f. fig. 2.8) accumu-lation, (b) ratio between corrected and uncorrected accumulation
2.3. Attenuation Fig. 2.10 shows a scatter plot of the uncorrected and the corrected data. It can be seen that the method effectively corrects the larger precipitation amounts towards a much better agreement between radar and gauge data. On the other hand, a tendency to overcorrect the small to medium intensity rainfalls is also apparent.
0 10 20 30 40 50 60
gauge values [mm]
0 10 20 30 40 50 60
radar values [mm]
uncorrected corrected
Figure 2.10.:Scatterplot of hourly accumulations for data corrected for clutter only (un-corrected) and corrected for clutter and attenuation ((un-corrected), Radar T¨urkheim, May-July 2008
Tab. 2.4 gives some summary statistics based on the data presented in fig. 2.10.
Both error measures slightly increase for the attenuation corrected data set in ad-dition to a change in bias from underestimation to overestimation. The successful correction of the largest measurements is most likely the reason for the improved slope and correlation values.
Table 2.4.: Aggregate error measures for attenuation correction; agreement between hourly radar and gauge accumulations
RMSE MAE bias slope correlation uncorrected 0.463 0.061 0.859 0.531 0.774 corrected 0.485 0.075 1.238 0.780 0.798
To further assess the corrective performance of the method, fig. 2.11 shows the distribution of several error and goodness-of-fit measures. Figs. 2.11 (a) and (b), show the distribution of RMSE and MAE, with RMSE being chosen as a standard measure of error and MAE because it is a bit less susceptible to outliers than RMSE.
For figs. 2.11 (c) and (d) the difference between corrected and uncorrected RMSE and MAE, respectively, was taken and again the resulting distribution was plotted.
In panels (c) through (f) of fig. 2.11 a negative value implies an improvement, while a positive value means that the ’correction’ actually makes things worse. Fig. 2.11c shows that improvement only for roughly a third of the stations, while there is no improvement for the others. The MAE-difference in fig. 2.11d only shows improve-ment for one station. This is most probably due to the greater relative weight that many small to medium deviations get during the calculation of MAE.
Two additional measures of agreement were calculated and are shown in figs. 2.11 (e) and (f). Bias was calculated as the ratio between the mean of the radar data and the mean of the gauge data. Slope represents the slope of a linear regression between gauge and radar data, with the gauge data being the independent variable.
The change between uncorrected and corrected radar data was calculated according to
δ=|1−bc| − |1−bu| (2.19) with bc being the value for the corrected and bu the respective value for the uncor-rected data. As both bias and slope should be 1, eqn. 2.19 shows an improvement as a negative value and a worsening as a positive value. It does not make a statement whether an underestimation is still an underestimation or whether there has been a more profound change to the dependence. However, this can be assessed using the other measures and the scatterplot.
Together with MAE-difference, the change in Bias, which also worsened for about two thirds of the data this shows that there is overcorrection in the majority of cases. The major improvements in terms of slope change, may on the other hand be attributed to the successful correction at larger intensities.
In summary, the attenuation correction has shown its potential in correcting in-tense precipitation, while the constraints imposed on the correction prevented large overcorrections. However, the method does overcorrect for small to medium intensi-ties, leading to increased absolute errors and biases as compared to uncorrected radar data. Further refinements of the method are possible and should focus on removing these biases.
2.3. Attenuation
0.2 0.4 0.6 0.8 1.0 1.2
RMSE [mm]
0 2 4 6 8 10 12 14 16 18
abs. frequency [-]
(a)RMSE
0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 MAE [mm]
0 1 2 3 4 5 6 7 8
abs. frequency [-]
(b)MAE
0.4 0.2 0.0 0.2 0.4
RMSE difference [mm]
0 5 10 15 20
abs. frequency [-]
(c) RMSE-difference
0.04 0.02 0.00 0.02 0.04 MAE difference [mm]
0 2 4 6 8 10 12 14 16
abs. frequency [-]
(d) MAE-difference
0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 Bias change [-]
0 2 4 6 8 10
abs. frequency [-]
(e) Bias
0.4 0.2 0.0 0.2 0.4
Slope change [-]
0 2 4 6 8 10 12 14
abs. frequency [-]
(f ) Slope
Figure 2.11.:Distribution different error measures and their changes due to attenuation correction